I'm not really big on cats, and especially not cat Pokemon. I can't think of any of them that appeal to me off the top of my head, TBH.
I am a dog person, and like most dog Pokemon. Houndoom is probably my favorite. Growlithe, Lillipup/Herdier, Furfrou, and Rockruff/Lycanroc (excluding the Midnight form, which I don't care for) are all great, as well.
Mew, though it also has mouse-like features, is probably my favorite cat-like Pokémon. It's my favorite type of Pokémon card to collect (I'm up to 28 right now). Legendaries/Mythicals aside, I'm torn between Meowth and Persian. Meowth is more nostalgic (I love the my chrome Meowth TOPPS character card and the TCG GB promo), but I'd be more inclined to use the elegant Persian in-game or have one IRL.
Flareon is probably my favorite dog-like Pokémon at the moment. I randomly found a full-odds Shiny Eevee during my second playthrough of X while hunting for rain to evolve my Sliggoo before facing the E4. Although I didn't consider Flareon my favorite Eeveelution at the time, I chose it since I already had most of the others via Friend Safari or Masuda method from previous files. Now Chipotle is one of my most-prized Shiny Pokémon.
As a cat person though, I prefer thinking of Flareon as a fantasy creature/spirit than equating it to a dog or any other real-world animal. Its Jungle artwork, my second favorite artwork in the entire TCG, sort of gives me that vibe—it looks like something that would come from a Cardcaptor's Clow Card.
I've always been really fond of the Skitty line, still am, honestly. It has a good ability in Normalize (every move Skitty does gives a STAB bonus), as for dogs... hmm, Growlithe & Rockruff are pretty cool. There's really too many dog Pokemon to choose from.