Maybe Dogz?

https://en.wikipedia.org/wiki/Dogz_and_Catz

There was at least one Tamagotchi game.

https://www.mobygames.com/game/2862/tamagotchi-cd-rom/
https://www.youtube.com/watch?v=bYJLt2cf5og

I'm after something I can download for free, but I think I like the idea of Tamigotchi. 😀

Neko was another desktop pet cat, there is one called Neko98 that might work.
https://archive.org/details/neko98

eSheep is said to work on Win98 (and later), but the free version link seems to be a reworked 64bit version only:
https://archive.org/details/sheep_202205
https://toolslib.net/downloads/viewdownload/183-esheep/

The original (called Sheep, by Empire Interactive) can be found online also, but is not free, so I won't link it.

I just downloaded Neko and think I have eSheep. Should I look for him/her in my downloads folder?

...probably, if it was there you put him?
(I've never used these on the PC what I can recall, but I had Neko on a Sun workstation)

Well, he didn't come out to play. Maybe I should. 😀 Will Catz and Neko work on Win11?

No idea, I use neither Win10 nor 11.

Okay. Thank you. 😀

I found Catz 3 the other day on a magazine cover disk. Installed it on my Pentium MMX machine and my 10yr old daughter has been having a great time with it

I'm glad he made her happy! 😀 If you also have a more modern computer, I recommend Worms Deskmate. He used to crawl all over my computer screen, do tricks and talk, but I stopped letting him come out to play. 😀 I used to enjoy him, though.

Good news! He's now playing on my desktop. 😀