No need to change partners just because someone else have the same one. That would cut clubs down to quite a few members then :p
What cat/dog is the most useful?
For cats in my opinion, I'd have to say Luxray. It's a great Electric type in battle, and out of all the cat Pokémon, I just like it the most, both design wise and battle wise, though design doesn't really affect how useful it is. But yeah, it's a pretty great teammate, and I've used it everytime I've gone through Sinnoh.
For dogs, definitely Arcanine. It's super fast, super strong, and just over-all great. It was one of my favourites back in the day, and I still like it a lot. Definitely a useful Pokémon, though I guess Lucario would come pretty close. I guess the topic is a bit more factual though rather than opinionated, but I can't really help but being biased, ehe.
Which one is most useless?
For cats, definitely Meowth. It isn't really useful for much except Payday, which I have never even used, and it isn't really known for the one that spams Attract like Skitty and Delcatty, so yeah. Out of the lot, it's the most boring and useless, I'd say.
For dogs, I'd say either Lillipup or Snubull. Lillipup is probably the weakest of the two, but it kinda makes up for that by evolving into some pretty decently strong dogs, but at its stage it just isn't really that useful. Snubull just never appealed to me, I didn't use it for long because I found it boring, and well. That's pretty much all I can say. So I'd probably have to go with Snubull as the most useless.