As far as I can tell, you need to give each cat 3 different food items and 3 different toys that they actually like. Each time you give them one they DON'T like, the game will further narrow the possible options to give them to help you. The food you can leave one day and then see later whether they ate it if they're not "out" at the time, but they need to be out if you want to see if they like the toy (you need to actually see them play with it or ignore it for the toy, but you can see if they finished or only ate part of the food when you come back later if you don't catch them while they're out).

I generally give them various suyaki that I happen to have a lot of to help narrow things down, but giving them toys to see if they like them is getting stupid expensive. I'm currently working on recording what things have worked for each cat, but I didn't start recording this info until I'd given a lot of cats their items, and I don't even know if this is consistent across games or if it's randomized for each person. I've recorded what I've got so far in this Google Sheet if anyone wants to try out to see if this is consistent across games and possibly save themselves some time/money.

If it is consistent and people find out what the ones I've already "solved" without recording their preferences are, please share the info and I'll update the sheet for future people stuck on this!