Why does AIM need my phone number? And why can't it find it?
Apparently this is by design (did not know that until today). You can only use the AIM, MSN, and Yahoo! messengers on the G1 if you're using a T-Mobile SIM card. I suppose this may apply to other unlocked phones since the comment on the bug tracker implies that it's actually got something to do with the way Android or its APIs are doing things under the hood.

I'd just recommend trying a different client, I guess. I personally recommend Trillian or Meebo, but there's a bunch of others. Trillian is $5 (or you can go get an old beta version - they were free), but I like the UI and features a bit more. Meebo is free but also very good. One big difference is that Trillian supports C2DM, but that doesn't really apply here since it wasn't introduced until Android 2.2

As for the "why": I'm not sure, really. Matt may be correct, but AIM can send SMS messages from desktop clients, so your number seems like it might not really be necessary. I'm inclined to agree with that guess, though, since I can't think of any other reasons.