Finder will show the home folder as a home symbol in some contexts (left nav bar, particularly), but it is an ordinary folder on macOS. And Finder is not particularly integrated with zsh or any other shell. Not past double-clicking on certain-named shell scripts to execute them, or dragging files or folder icons into Terminal.app to transfer the path, that is. Some zsh doc will show Linux or other systems’ icons, as well.
I’ve never bothered to specifically run any initialization scripts for zsh (and my home folder is, well, ancient), only adding the zsh “dot” (“hidden”) scripts (.zshrc, .zprofile, etc) and zsh settings as needed. In macOS Finder, use the Command-Shift-Period toggle to show or hide these and other hidden files.
If you want to see what that script does, here it is:
https://github.com/zsh-users/zsh/blob/master/Functions/Newuser/zsh-newuser-install
You can test that zsh install script with a newly-created macOS login. After creating the new login, login to it and open Terminal.app, and then invoke the script there.then have a look at the generated .zshrc script. I would not expect that to then alter what Finder shows, either.
Given it only claims to modify .zshrc, after testing in a throw-away login, you can move your .zshrc and related files aside, run it, and merge the results as needed.
Some related reading: https://apple.stackexchange.com/a/361957