If you put 20 mice in a bag they would just thrash around loudly untill the backpack tears and they spill out.
It seems to me that the player just wants to make sure that the cat has kitty rations so it doesn't starve. I think you should reassure the player that the cat won't starve and you'll just hand wave it away and just assume that the cat can finds its own food and that one of his rations feeds him and his cat. And that the cat jumps in and out of the backpack at its own leisure and isn't just stuffed in the pack, which is what I think the Player had in mind.
I think you should treat the cat like a Character Trait instead of an NPC.
The cat doesn't have any meaningful stats and it doesn't mechanically do anything.
You never have to keep track of it. It's there when you need him, and not there when you don't.
Just treat the cat as something for the PCs and NPCs to roleplay off of, and as flavour/fluff in the fiction, and as an extension of the PC.
Like when the player succeeds on a Perception Check. "You notice Coconut (Yes I already gave a name to your players cat, she's also a she and her fur is chocolate brown with a white muzzle, star, and the tips of her fuzzy ears.) turn her head towards a bush and hiss. You follow her gaze and notice a Wolf in the bushes."
Or when the player fails at a Charisma Check. "You tried to seduce the Barmaid but she sneezes in your face, gives an angry look at Coconut and says shes deathly allergic to cats and stomps away."
Or, Orc misses attack. "The Orc was about to bring down his axe on you but Coconut leaps out of your backpack right into his face causing him to miss, then she leaps off and scampers off to safety."
Don't let Coconut get lost or hurt. If a fireball is dropped in the middle of the party, Coconut was in the backpack safe and sound, or ran into the bushes in time. When a fight starts, she runs off. And when it ends, she scampers right back.
But! If the Player happens to be the type that gives big reactions to things then I think it would be fun to FEIGN peril for Coconut. "You all just barely escaped the collapsing tunnel. But where is Coconut? you don't see her, and you look back at the section of the tunnel that collapsed and your stomach sinks... (Let them think the worse for a few beats.) ... Then you hear a meow further up the tunnel. She was perfectly safe the whole time."
I think pets can be a lot of fun if used right. But if you have to babysit them and constantly worry that they might die or be used against you, then you can end up regret it. It's why PCs never have any family, because they don't want the DM to put them in danger or use them against the players PC.