I'm sorry but games do not attach to your account like origin or steam. Yes, the serial code attachs to uplay but you cant install the game from uplay, you have to download the game from gamersgate/steam.
The only way you can download the game from uplay is if you bought the game from uplay client itself or ubisoft store page. If you dont believe me uninstall your game from steam or gamersgate and it will dissapear from uplay client.
Btw I tested this because it didn't sound right. It is absolutely untrue. Stam copies of the game do function how you mentioned.
However I bought assassins creed 2 from gamers gate. I installed it and it launched uplay with all my other uplay games and asked for a serial number. I put it in and the game worked great and shows up next to titles purchased from uplay directly. So I went a step further and uninstalled the game. I closed uplay and reopened it and the game was till there and had a download tab. I clicked the download button and it started downloading.