Dear Mr. Price
I'm not entirely sure if the problem with "pet empathy" in video games is a problem with the game industry or with you. Don't be offended, but I suspect the problem is with you. Either you have become too old and jaded in your emotions, or perhaps you simply pick the wrong games to play if that is what your looking for. Pokemon for example is a horrendously bad example for someone looking toward games that highlight a relationship between master and pet.
I'd point a finger at a rather unique survival horror title called "Rule Of Rose" for an example of what your talking about. To me at least that is probably the best relationship between master and dog done in video games, especially when you get to the end of the game and more or less figure out what was going on. All I can say is wow... just wow.
There have been others of course though none that were that good. "Haunting Ground" comes to mind, and then of course there is Fable II, though I must admit that the pet in that game needed some work, the final scene involving the dog and the desician you have to make involiving it hit a lot of people fairly hard irregardless of what they say. I was there for some of the initial discussions on that one when the game was brand new.
Then, while it's arguably a bad example, there is Dogmeat. Dogmeat in Fallout is sort of a Phenomena in of himself. Oh sure he lacks a lot of the antics/writing of other pets, but consider that people who play Fallout 3 were immediatly screaming "where is Dogmeat" since he was hidden in the Scrapyard, and then once they have him they go to crazy lengths to protect the little furball, ranging from keeping him in their house (semi-defeating the purpose) or installing one of the ever-popular "put Dogmeat in God Mode" mods (PC version) so they can keep him around. I wouldn't be surprised if the most popular use for the PC cheat console is actually to respawn Dogmeat when he glitches out hunting items.
My point is that I think the relationship between master and pet has been captured pretty well in video games, albeit not in many RPGS based around capturing pets largely because there is typically such a huge bestiary that your constantly swapping them out, and objectives that require you to switch things up as opposed to keeping your favorite pet handy. Plus in a game like say Pokemon they really don't do much except fight for you when you pop them out of balls. They generally don't walk around with you like in some of the other games mentioned above.
Generally speaking I think you don't see more of it because the games that really made an effort were not smash successes (except maybe Fable II if you look purely at sales, but that is the weakest example of the ones I listed above I think).
At any rate, anyone who is looking for a relationship between pet and master should probably check out Rule Of Rose. Not only is it a very unique and atmospheric game that deserves more attention, but it should give you pretty much everything your looking for and perhaps more. This is simply a personal opinion though.
Now, all we need is an equivilent for cat people.