Which tool is this about? Where is its repository?

https://github.com/zdharma-continuum/zinit
It's also at
https://github.com/psprint/Zinit-4 what's interesting is this one points to zdharma-continuum if you don't use the appimage
And a version that the dev doesn't support
https://github.com/z-shell/zi

Which operating systems are supported by this tool?

I believe everything with zsh but I know linux is it's main support focus

Which exact commands should Topgrade run?

zinit update --parallel
you can also do
zinit update --parallel 40 to make it faster

I want to suggest some general feature

Topgrade should...
Make zinit use 'zinit update --parallel'