If I was to create a pet site, I would make features with two things in mind. 1) They have to be fun. 2) They have to be related to pets.
Of course adding other features that don't fit into both criteria is always nice. For example, the feeding and playing with your pets features I've seen have always been cute and a nice thing on the side, they are obvious pet features, but they aren't really fun (at least the ones I've used, they just felt kinda repetative and mundane). Or an example of something fun but not pet related necessarily I would say might be battling. Great feature to have, but we don't fight pets in real life, it has more of a Pokemon feel (so I wouldn't center a petsite around it if I wanted to focus on pets).
An example of something that's both fun and pet related might be adoptions and adoption agencies. I remember on Neopets, people loved making these. Theyd collect a bunch of unwanted pets from the pound and rehome them. This is a totally believable scenario for pets in real life, and also is a lot of fun. A whole site could probably be based off a pet adoption concept, but if not, at least it could be one of the core features. Another core feature could be pet shows (like you train them to do tricks and such and compete with other players, not like shows for breeders like in SIMs, but pet shows like that TV program Pet Star).
Again, don't get me wrong. Totally love battling, playing with your pets, etc. and sites like Subeta I would still consider a pet site, this is just what I would do personally if I was going to focus on the pet aspect of a pet site.
This is just my opinion and some brainstorming though.