I know (quite well) a dog who bit a young child (2 or 3) for either hugging her or trying to pick her up. Don't know for sure which because the adult "supervising" the interaction wasn't actually looking at them when it happened, but I had previously told the child they couldn't pick my dogs up (Bus is fine with anything, Annie at the time wasn't tolerant of manhandling), and also had to tell her to not kick my dogs, so I would suspect she may have done something similar to provoke the bite the other dog gave. The dog was otherwise fine with kids before, and still seems fine with kids, though it's not allowed around young kids anymore for obvious reasons- low pain/fright threshold and willingness to use her teeth to tell someone to back off= not a good match for unpredictable young children.
I know lots of dogs who seem indifferent to hugging.... they don't really seem to "enjoy" it, but they tolerate it with good nature, like baths, dressing up, and all the other weird stuff people do to them. Then there are dogs that even the owners can't hug, because for whatever reason the dog freaks out. At the other end of the spectrum are dogs that really like to be hugged, and some of them even initiate it!
My APBT, Haley, lived to be on people, next to people, under people, basically touching you as much and often as possible, like she was trying to share your skin. She would plop onto a stranger's lap, and scoot up them until she was practically in their arms. Like this:
If I went somewhere in the car as a passenger and had her, she would sometimes climb into the seat, then sit up on her haunches and put her leg (she only had one front) on my shoulder, then rest her head on the other shoulder, literally trying to hug me- I hated it on hot days, and would try to shake her off me (to no avail usually), but it was nice when it was cold out. She'd take a hug from anyone, anytime
Funny enough, she frequently did the "paw over the back/leaning" to other dogs to try to pick a fight, except she used the "stump" of her amputated leg. She only ever "stumped" people trying to get their attention (woke me up more than once!), never in a conflict seeking fashion. Similar type of contact, but entirely different contexts and behavior.
Of my current dogs, Bus is indifferent to hugs mostly, they don't bother him, and sometimes he'll "hug" us by climbing up and leaning his head/neck against our neck/face (and putting his cold, wet nose on us!) if he wants our attention/affection. Strangers can hug him (he's a therapy dog, so it's his job to tolerate and even like all the weird stuff people do
) and he doesn't mind or appears to like it, but he usually only seeks out hugs from people he knows well. Annie wasn't much on hugs as a young dog, but has become more tolerant as an old fart, though she's so frail I wouldn't let strangers (especially kids) hug her for fear that they might hurt her. There's a good chance she would have bitten a stranger that snatched her up and hugged her as a younger dog- she has the typical aloof dachshund temperament, and was never keen on manhandling by strangers.
I think too that it depends on how you hug a dog- a face to face, two arms around the neck hug is about as direct as it can get, with little "escape route" for the dog to duck out if they feel uncomfortable. One arm slung over the dog while they sit beside you is probably more acceptable to dogs who may have some reservations, and if not, they have a much wider opening to make an exit if they choose.
