What exactly does `ssh -T user@domain verify` do?

Disable pseudo-terminal allocation

This means that the command verify is run directly on the other side and the standard input/output/error are connected directly to your local terminal.

For commands, there is usually no difference, but it is difference if you want to run interactive scripts (shell). In that point, your output would not be handled by remote terminal, but with your own and the output would not be so pretty. You can try with ssh -T other_host, where you have terminal access (you don't have on github). On github, it is also one of the resources-saving task, since TTY allocation is "not for free" if you have 31 million users (October 2018).

ssh -T [email protected] verify

This logs into the account [email protected] and executes the command verify there. So it definitely goes to the network. The command verify is specific to github and creates a token which you can use to prove to the support that you are the owner of a specific key used to log into this account.