Hi @dhmskrn!
WordPress unfortunately does not handle notifications in a consistent manner. Some default communications are handled by “pluggable” methods — methods you can replace wholesale with your own — some offer filters for overriding the content (but not the send method), and still others are hard-coded and offer no direct solutions for replacement (though if you’re determined enough, you could always implement a custom handling solution further up the chain, for example, a custom registration or password reset form, though care must be taken to ensure data is handled safely!).
Pluggable methods offer the most flexibility for overriding, but there are only a few of them:
wp_notify_postauthor(), wp_notify_moderator(), wp_new_user_notification(), and wp_password_change_notification().