(Damn, I had wanted to post this yesterday, but then I forgot to quite finish it and it ended up sitting in a tab, unsent.)
Yes, it does: it means that you cannot count on it as a canonical form of identity for any long period of time, as people expect to be able to change it; in fact, many people go through "the great purge" every couple years, deleting their e-mail address and selecting a new one, in order to purposely reboot the people who have their address: to them, it is a way to purposely restart their identity.
You thereby have to think of e-mail addresses as more akin to your home address. If you ask me to log in with my home address, yes, that works: it doesn't work for everyone, as not everyone in the world has a home address, but the same thing can be said for e-mail addresses. Sometimes people will share a home address, but surprise surprise: sometimes people share e-mail addresses as well.
When I am asked to log in to that form with my home address, and it works, you now might claim I've accepted it as part of my identity. Well, I haven't: I'm going to change my home address at some point, and someone else is then going to start living here, which is exactly what happens to many people who use ISP or University -provided e-mail. Hell, it also happens to people with vanity e-mail addresses if they let their domain registration expire (as happened to one of my friends, who otherwise was using the same e-mail address for a very very long period of time).
Yes: it works temporarily, but it isn't my identity, and eventually it will fail, and unless you are really really weird (like, you are the kind of technology person who would probably consider it digital suicide to allow their domain name to expire, and has had the same e-mail address now for well over a decade), it will fail sooner than later, and may even fail on purpose when users invalidate it.