Tascots is a nice modern desktop pet application that is in beta atm..!! It's accessible through the creator's Ko-Fi; a $3 support payment allows you to access the discord where you can download the program as a .zip file.
It would be a bit less accessible for sharing than shimeji--each Tascot character is saved in the .tascot proprietary filetype that the program reads to add them to your catalogue of spawnable characters. While it does not have any kind of DRM, the editor and main program are bundled into one, so I imagine the course of action if one were to share their creations on itch would be to post the .tascot file and simply encourage others to support/purchase the program to be able to use it ^_^ (as opposed to shimeji where folks freely redistribute the program itself alongside the character files).
It is very inexpensive though and pretty robust! It is very fun to mess with, definitely recommend.
Also JUST found an incredibly simple option, VirtualPeto that I am also looking into.. It looks like another simple shimeji type alternative, with a range of features as simple as Display A Gif On Your Screen up to having a little guy with several animation states that walks around and follows your mouse, and can have custom animations as well as eating or playing with a little toy.. It also seems to have some gif related tools built in and is free..!
Will keep updating if I come up with more options ^_^