Proton is an email service. They manage email for you, either with their domain, or one that you own.
You can’t simply transfer an email address that’s used with another email service (eg gmail) over to protonmail. It’s not your email address in the sense that you don’t own the domain and they could take it from you at any time.
If you’re looking for an email client, check out thunderbird.
If you’re really keen on doing this, you could have your own server and a script send these emails, but I assume that involves a lot of work.