When I'm logged in and browsing through games, I should be able to re-download a game I've already purchased, when I come across it. I shouldn't have to go through my emails or purchase history on itch.io. It would also be nice if there were notifications indicating that there are updates to games, since I've last downloaded them. Finally, developers should be required to assign version numbers to games that they publish, so we know what we already have or don't have.

Itch.io does show me a banner at the top of games that I've purchased or supported.  If you're not seeing that then it may be that it's a duplicate of the game or the developer deleted the game page and re-uploaded it - (they're not meant to do either of these two things).

Else it could be that you got a game at 100% off that wasn't claimable.

Under More information on the game pages - there are 2 dates - a date on which the game was published and a number of days since it was last updated.  Probably an update date would be more useful than the number of days - but you'd be able to tell from there if that's the version you have.