This is more of a mail server question than a next.js one.

You can self-host your own mail server on a relatively cheap VPS ($5/mo) on Hetzner with a docker container for either mailcow (if you want to be able to send and receive email) or postfix (if you only need to send out emails for auth, notifications etc.)

Another way is to use something like Resend which gets annoying by requiring you to go on a paid plan if you want to use it for 2 apps with separate domains.