How do I change the developer's email that a new user gets when they sign-up for my site? See image below..

Will I have to completely set up a new firebase account with a different email address and change all the code corresponding with this one? Or is there another way?
If I'm understanding you correctly, you want your project to be associated with a different email, correct? If that's what you mean, then you can go to the Firebase console, click the gear>Users and Permissions, add the other email as the owner, accept the role with the other email, and then delete the original email you no longer want to associate.
Email id and all the other information that is appearing under Developer info popup can be changed in OAuth consent screen.
Here is the path:
"https://console.cloud.google.com/" -> "APIs & Services" -> "Credentials" -> "OAuth consent screen"
Here is the screenshot:

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