My 3 dogs are Timothy Tiitus (named after 2 chapters in the bible), Bob (named after my uncle), and Diana (named after Princess Diana).

You ask, "CAN he call his dog Bob?" So the answer is, yes, of course he CAN. Is it COMMON to do so? No, it's not common. It's not a tradition, or anything like that. But some people do, and they have the right to do so. You mention you told your students about this. Well, make it clear that people CAN do so, but it is not a tradition to do so.

I've never come across anyone who's named an animal after a person. Animals are animals, after all, and it would strike me as strange and a bit insulting to name a pet after a person - or vice versa.

You can call your pet anything you want :). But I don't think I've ever seen that here in California. Yes, people give their pets people names some times (my cat's name is Polly, for example, though my other cat's name is Whisky); but I've never seen people named their pets after relatives or friends or anything like that. And if it actually does happen I really doubt it ONLY happens in English speaking countries.

My cats names are Tobi and Sushi :) Nobody I know has named their pets after their relatives. People here mainly name their children after beloved relatives.