The correct answer should’ve addressed your problem:
chsh: /usr/bin/zsh: non-standard shell
The reason this is the case is because chsh will only accept shells that are defined in the file /etc/shells, as you can see by reading the manual for chsh:
chsh will accept the full pathname of any executable file on
the system. However, it will issue a warning if the shell is not
listed in the
/etc/shells file.
To solve this problem and make zsh the default shell, you should thus:
$ sudo echo "$(which zsh)" >> /etc/shells
$ chsh -s $(which zsh)
Obviously, I assume that zsh is in your path here. This solution will also work if you, for example, choose to install the latest zsh with brew install zsh.
EDIT (thanks for ThisIsFlorianK for the comment):
Depending on your shell setup you may get a message saying /etc/shells: Permission denied. You can find information about this issue here.
To work around it, use the following instead:
$ sudo sh -c "echo $(which zsh) >> /etc/shells"
$ chsh -s $(which zsh)