Once it's set up, you can , and use your Google account to find, add, and manage those contacts.
It doesn't need to be your legal name, but it should be the way you want to be identified in Gmail.
So, if you are getting that option, you can create a Google account without entering any phone number.
Once there, click on the Forward a copy of incoming mail to button.
Follow the instructions in the Gmail app to create a new Google account.
Go to your Google profile icon in the top right-hand corner of the screen.