Why would you register with an e-mail address that's not yours in the first place?

They can't be, because Gmail obviously won't let you register an address that's already registered.

Yes, but how can that ever cause a problem to the end user?

You have to be a bit careful with that kind of argument though. According to the specs the local part of an email address (before the @) can also be case sensitive so me@example.com and ME@example.com could be different email addresses. In the early days of a new system we implemented account email addresses following what the rules really say and making no assumptions. We then spent way too much time dealing with users who signed up with one address but then tried to log in with a "different" one because they had caps lock on or some auto-capitalisation feature in whatever browser they were using. Lesson learned - users do assume those are all the same address and cleaning up a database that sometimes has accounts for both me@example.com and ME@example.com isn't much fun.

There is no RFC about how the local-part is interpreted: even the gmail +tag syntax is an extension some servers implement and some don't. The only way to accurately test equality of email addresses is to send an email to them.

Yes, I know, that's the whole point. The mistakes that most people (including yourself) are attributing to the dots, are actually usually caused by human carelessness -- spelling errors, mixing up gmail with hotmail, "dave" instead of "david" etc.