I came across a (ridiculous) question + answer purporting to be from myself while I was looking through 'questions from new users'. It was removed, but I think that this reveals a serious flaw in the system. Is it soon to be corrected?
In most cases, it is perfectly reasonable for many users to have the same name, so it doesn't make sense to force them to be unique. For example, I made a query to find users with the same display name, and there are 224 Alexes on the site. (I for one don't like having to add numbers to my display name; others probably feel the same way.)
However, it is not ok to impersonate other users. If you think you see someone impersonating another user, flag for moderator attention with a message like this:
This user appears to be impersonating [this user](link to profile)
Or in your case, you can say "impersonating me".
If the user has copied more than just your username (e.g. profile picture, bio text), be sure to mention this in your flag. There's a big difference between two people being called Alex and two Alexes with the same profile picture and the exact same bio.
In your case, the user posted nothing but trollish content, which was flagged as abusive and deleted. That, combined with the sheer unusualness of your name, seems to have led moderators to delete the user.
See also Tim Post's answer here:
Accounts aren't tied to display names, they aren't required to be unique. However, if you participate in comments or chat with someone that has the same display name, you'll get a bit of noise in your global inbox if someone mentions one of you.
What you can't do is impersonate someone else, or otherwise harass people:
Under no circumstances will Subscriber use the Network or the Service to (a) send unsolicited e-mails, bulk mail, spam or other materials to users of the Network or any other individual, (b) harass, threaten, stalk or abuse any person or party, including other users of the Network, (c) create a false identity or to impersonate another person, or (d) knowingly post any false, inaccurate or incomplete material.
There's a difference between sharing the same name as another user, and pretending to be another user.
Barring that, there's no issue if several or more people share the same display name.