Does it feel odd to anyone else that you click the "Request an invite" button and it doesn't ask who you are? Also nowhere on the page does it show my identity.
It was odd indeed. And I checked to see which account I was logged in under, and it wasn't my "normal" account. So I switched to my Gmail/Google+ account and re-clicked, and then it said "you've already requested an invite." Odd.
They're probably setting a cookie, did you try clearing them or using a different browser?
Perhaps if you don't realize you're logged into gmail. I thought it was well understood that being logged into gmail meant you were logged into all of google's services?
I am logged into 2 Gmail accounts at the moment, which one did it connect with?
You'll find out when you get an e-mail to one of the accounts.
It asks me to sign in when I click "Request Invite" in a browser that has never logged in to google. Also asks me to sign in if I've signed out of google.