Email addresses need the same regulation. The arguments that lead to phone number portability apply to email addresses as well. And I would argue that email addresses are even more important than phone numbers at this point (it's the single key to all online accounts, most bills, documents, statements are emailed as PDFs, a lot of government services expect a working email address).
Email addresses have become critical and portability needs to become a requirement for all email services. There are technical issues, for example if someone cancels Gmail service, how can the @gmail.com address be moved elsewhere? It's not as simple as phone number portability. Maybe a regulation that any email service must provide forwarding service to another email address even if the service is no longer active? Or maybe a trusted mapping that exists outside any single service, kind of like a DNS for email addresses.