Yes you are right there are a lot of games out there that were pet related

Some that I know of are...

Roland's Rat Race - you took control of international mega-superstar Roland Rat and his friends - Kevin the Gerbil, Errol the Hamster, etc... Roland Rat is a puppet rat that had his main highs in the 80s on TV-AM, British TV morning show.

Monty Mole - I presume you've already got this one, on the Spectrum there was a massive collection of Monty Mole games, one of the many platformers similar to Jet Set Willy, but instead of Miner Willy it featured Monty Mole - a mole - which you controlled.

Sim Ant and Sim Life - both from Maxis, one you are an ant and you have to take over an entire garden, and eventually make moves on the house, Sim Life was a simulation of life studying things like biodiversity, what happens when you introduce alien species into a colony, and how to either nurture and grow a sustainable eco-system or destroy it by introducing the wrong predators

Zenobi software also did a number of titles such as Balrog & The Cat, Behind Closed Doors, etc all of these were text based adventure games, most of them featured throughout some kind of interaction in the quests with some form of animal, normally a cat called Zenobi, which was the actual cat owned by the guy who ran Zenobi software (he still runs Zenobi software to this day, but sadly the cat is no longer around!). Quite often typing in messages in the game like Zenobi would receive a text response like Miaow! or some obscenities may produce other messages like I would but the cat is looking at me strangely.

I'm sure there were quite a few old games were you controlled an animal but I can't think of too many that would be on the lines of taking care of a pet.

The closest thing I can maybe think of playing that fits would be a creature battle game for the C64 called Mail Order Monsters. You started out selecting one of a number of base monster types, go do battle against a computer opponent or I think one of 3 missions and earned points to put towards upgrades. And repeat. I think I also remember there was some sort of multi-player ability where you could load monster's you've saved to disk to battle head to head against your friend's monsters.

Another animal-centric game (not really a pet game though) that I really liked and, am a bit surprised I haven't seen remakes attempted, was Frog Bog for the Intelevision. Designed as a head to head two player game, you play as a frog, there are two lily pads you can jump between, and the goal is simply to eat as many of the bugs flying around as you can before nightfall and score more points than your opponent. If you're playing a low difficulty all you have to do is time your jumps right and you would jump at a consistent height, arc, and direction. Higher difficulty gave you more control and a higher chance of jumping off the lily pad which cost you time as your frog would swim back to the pad.