Welcome to the Auth0 Community!
I’ve tested a user’s login with both username and email, upon checking the Management API’s Get User by ID, I’ve found that Identifier will return the type an value used on each login, which is found under the details object.
You should be able to invoke the Management API within Actions; for additional steps on how to perform this, please take a look at our Knowledge Article.
Let me know if this is what you were looking for.
Have a great one!
Gerald