I've had exactly one pet (once past the training quests). If I dismiss him and try to train another, game says "You already have a pet." I haven't tried putting him in the kennel, yet, maybe that's the answer.
But even with the kennel, seems like the maximum number of pets anyone can have is three -- two in the kennel and one at work with you. Is that right?
Or, if I want to get rid of him forever, I suppose I could ust let him get killed or starve him till he leaves me? Will I lose the pet abilities he knows in those cases.
Pretty much, I'm willing to stick with the known, but it seems worthwhile to get a few more pets to get Claw and Bite and so on. I'd just like to avoid doing something really stupid and losing my 'best friend' forever :)