We humans love to philosophise about dogs. Are they still basically wolves? Do they know we're not dogs? How does that play into how we interact with them?

Let's go back to where it all started: wolves. Or rather, the process of domestication. Diamond's six criteria for domestication are diet, growth rate, disposition, willingness to breed in captivity, flexible social hierarchy, and panic. From his paper in the prestigious journal Nature, Evolution, consequences and future of plant and animal domestication, 2002:

Among wild mammal species that were never domesticated, the six main obstacles proved to be a diet not easily supplied by humans (hence no domestic anteaters), slow growth rate and long birth spacing (for example, elephants and gorillas), nasty disposition (grizzly bears and rhinoceroses), reluctance to breed in captivity (pandas and cheetahs), lack of follow-the-leader dominance hierarchies (bighorn sheep and antelope), and tendency to panic in enclosures or when faced with predators (gazelles and deer, except reindeer).

A simpler article that covers domestication is here on Howstuffworks.

Of course the relevant part to this discussion is the flexible social hierarchy. Animals who can be domesticated are all group-living animals with a social structure in which they follow a leader--and they are willing to let humans step into that position of "chief" and direct them. Arguably, what kind of animal they perceive us as, is less relevant than the position they see us in.

In the end, what matters is RESULTS--what is the most functional way to interact with dogs, that benefits both us and our dogs? From that perspective, while it's very likely that dogs "know we are not dogs" (!), they have a fairly simple worldview. When they are stressed, they growl--whether it's at another dog, a person, or the family cat. Growling is what they've got. When another dog leans over their neck and shoulders, they know he's asserting himself--and a high ranking dog may not tolerate it. He *also* may not tolerate it from your nephew, and respond accordingly. There is a laundry list of behaviours that socially dominant dogs won't accept from subordinates, regardless of what species does them: standing over/hugging, waking a sleeping dog, guarding or withholding food, or direct stares are just a few.
Some basic temperament tests done on dogs are to stand above him and put your hands on his hackles, pick him up from above in a bear hug, or place your hand over his foreleg when he is lying down. (Pups are routinely held still, or turned upside-down.) Many dogs don't care one whit, because they're not competitive/assertive/dominant/choose your word. Some dogs DO care, and will react badly to this regardless of the offending species. (Some won't even take a biscuit if you've licked it first!) Still other dogs pay close attention to hierarchy, and will allow a "boss" dog, or a specific human, or maybe ALL humans, to do those things, but will not allow it from a subordinate. (It's also interesting to note a study showing that dominant dogs will not learn from a subordinate--but that subordinates will learn from the leader! Link to that scientific study )
The body language and signals that explain all this are fascinating and are out there for the taking, if you want to be an exceptional dog handler, but they're a bit off-topic here. Suffice it to say, a lean-over is a lean-over, a stare is a stare, and controlling space is perceived as just that--regardless of who does it.

While it IS a major factor in behaviour, the operant conditioning we lean on so heavily today is reductionist and incomplete for humans, and is equally incomplete for dogs. Dogs have rich and complex inner lives, social rules, & language (what we call "culture", in humans). Adding to that, "absolute science based methods" tend to be less effective when you're not in a laboratory. Strict behaviourism was a disaster for behaviourist founder Watson's children, & today contributes to the deaths of many dogs as well. Any intelligent animal requires a more wholistic understanding of their culture.

For more thoughts on the inadequacy of restricting our interaction with dogs to operant conditioning alone, with no consideration for our place in their social lives, you could read this piece from Temple Grandin, who has some terrific books and an excellent grasp of animals' inner worlds. An excerpt:

Rats and other animals can be trained to perform a complex sequence of behaviors by chaining together a series of simple operant responses. Skinner believed that even the most complex behaviors can be explained as a series of conditioned responses.
However, a rat's behavior is very limited in a Skinner box. It's a world with very little variation, and the rat has little opportunity to use its natural behaviors. It simply learns to push a lever to obtain food or prevent a shock. Skinnerian principles explain why a rat behaves a certain way in the sterile confines of a 30 x 30-cm Plexiglas box, but they don't reveal much about the behavior of a rat in the local dump. Outside of the laboratory, a rat's behavior is more complex.

...after training more than 6000 animals as diverse as reindeer, cockatoos, raccoons, porpoises, and whales for exhibition in zoos, natural history museums, department store displays, fair and trade convention exhibits, and television, the Brelands wrote a second article featured in the American Psychologist (1961), which stated, "our backgrounds in behaviorism had not prepared us for the shock of some of our failures."

While Skinner and his fellow Americans were refining the principles of operant conditioning on thousands of rats and mice, ethology was being developed in Europe. Ethology is the study of animal behavior in natural environments and the primary concern of the ethologists is instinctive or innate behavior (Eibl-Eibesfeldt and Kramer, 1958).
Understanding the mechanisms and programming that result in innate behavioral patterns and the motivations behind why animals behave the way they do is the primary focus of ethologists. Konrad Lorenz (1939, 1965, 1981) and Niko Tinbergen (1948, 1951) cataloged the behavior of many animals in their natural environments. Together they developed the ethogram. An ethogram is a complete listing of all the behaviors that an animal performs in its natural environment. The ethogram includes both innate and learned behaviors.

Today, scientists recognize the contributions of both the Skinnerian and the ethologists approach to understanding behavior.

Ethology is ALSO a science, and a very useful one if you want to get the most out of your relationship with your dog. :-)

Certainly, dogs can meet us partway! They can read our faces and understand our emotions. They often forgive our failings in understanding their language, to the best of their abilities--although it's fair to note that a wolf tends to be less forgiving than the average domestic dog, suggesting that learning the limitations of humans is something that took time to develop. (!) However, we--with our big brains, and our decision to incorporate dogs into our very foreign worlds-- are also obliged to meet them partway. While dogs can understand some human-speak, it is arguably arrogant and dishonorable for humans to refuse to learn to speak their language as well. Anyone who does so, quickly finds that dogs understand us just fine when we choose to use the language and social rules of their kind.

Animals of many varieties use their species' pacifying behaviours towards other species. Dogs lick, whine, give paw, offer airplane ears, and so on to humans even though they probably "know that humans are not dogs". They submissive-urinate for us, which is a behaviour that signifies their submission to other dogs.
When dogs attempt a corrective bite towards humans, they go for faces and hands, not for the hamstring bite of a fearful dog. However, when they feel threatened, they do indeed hamstring. They are still being dogs, because dogs are all that they CAN be.
Sheep dogs are accepted into sheep flocks, and treat the sheep as their pack.
Your dog may leave YOUR cats alone (they are part of his social group) but then attack a strange cat.
Can you think of other ways dogs treat other species--including humans--the same way they treat fellow dogs?

Ever since olden times, people who have studied animals intensely have noticed their ability to substitute humans into their worlds. From "The Intelligence of Animals", 1869 ... "Cuvier has devoted himself to the study of these societies. He follows the progress of the animal which born in the midst of the flock, is there developed and which at each epoch of its life learns from all which surrounds it to place its new existence in harmony with that of the old ones. The feebleness of the young animals is the cause of their obedience to the old, which possess strength, and the habit of obeying once adopted by the young is the reason why the power still remains with the most aged, although he has become in turn the most feeble.

According to Cuvier, he looks upon man as the leader of the flock. Man, says M Flourens, is to the animals only a member of their society. All his art is reduced to making himself acceptable to them as an associate, for let him once become their associate he soon becomes their chief, being superior to them in intelligence. Man does not therefore change the natural state of these animals, as Buffon says- on the contrary, he profits by it. In other terms having found the animals sociable, he renders them domesticated and thus domestication is not a singular case but a modification a natural consequence of animal sociability. Nearly all our domestic animals are naturally sociable. The goat, pig, dog, rabbit, all live by nature in society that is in herds or flocks. The cat is not really a domesticated animal; it is not subdued, only tamed--in the same way the bear lion and tiger even might be tamed, but not domesticated. Man's influence will make a sociable animal domesticated, but a solitary animal he can only tame.
M Flourens attributes the domesticity of animals to their social instincts. But whence comes this instinct? By what is it determined? Evidently by the organisation..."

From "Principles of Geology", 1830-33: An animal in domesticity, says M. F. Cuvier, is not essentially in a different situation, in regard to the feeling of restraint, from one left to itself. It lives in society without constraint, because, without doubt, it was a social animal; and it conforms itself to the will of man, because it had a chief, to which, in a wild state, it would have yielded obedience. There is nothing in its new situation that is not conformable to its propensities; it is satisfying its wants by submission to a master, and makes no sacrifice of its natural inclinations. All the social animals, when left to themselves, form herds more or less numerous; and all the individuals of the same herd know each other, are mutually attached, and will not allow a strange individual to join them. In a wild state, moreover, they obey some individual, which, by its superiority, has become the chief of the herd. Our domestic species had, originally, this sociability of disposition; and no solitary species, however easy it may be to tame it, has yet afforded true domestic races. We merely, therefore, develope, to our own advantage, propensities which propel the individuals of certain species to draw near to their fellows. The sheep which we have reared is induced to follow us, as it would be led to follow the flock among which it was brought up; and, when individuals of gregarious species have been accustomed to one master, it is he alone whom they acknowledge as their chief—he only whom they obey. “The elephant allows himself to be directed only by the carnac whom he has adopted; the dog itself, reared in solitude with its master, manifests a hostile disposition towards all others; and every body knows how dangerous it is to be in the midst of a herd of cows, in pasturages that are little frequented, when they have not at their head the keeper who takes care of them. “Every thing, therefore, tends to convince us, that formerly men were only with regard to the domestic animals, what those who are particularly charged with the care of them still are—namely, members of the society which these animals form among themselves; and, that they are only distinguished, in the general mass, by the authority which they have been enabled to assume from their superiority of intellect. Thus, every social animal which recognizes man as a member, and as the chief of its herd, is a domestic animal. It might even be said, that, from the moment when such an animal admits man as a member of its society, it is domesticated, as man could not enter into such society without becoming the chief of it.”

Lots of food for thought. So, do dogs "know we're not big weird dogs"? I suspect they do. Does it matter? Probably not...not in any practical sense.

Related bonus link: Intrinsic vs external motivators--why your dog wants to work for you based on your relationship, even more so than for that cookie.