Arguable email addresses now are more important, that phone numbers. Mobile carriers are legally required to allow you to port numbers. We need a legal framework that allows to have inalienable email addresses.
Due to the decentralized nature of email, you can't have inalienable email addresses except within a domain. But you can register your own domain, and then, it is tied to your real identity and you can transfer it between registrars, which may be closer to what you had in mind. Most registrars provide an email relay, so that's probably the best you can get.