Brotherhood is a full game, not a side story. Apparently when AC II was going through the creative process, the developers were in Italy and they intended on making Rome a part of AC II, but realized how huge it was not only in actual size, but for what they wanted in AC II. So they decided to focus on Rome in a separate game.
So yeah, it is absolutely a full game.
Also, I do not think Ezio was put into legendary assassin immediately. He actually started out with little fighting and assassination skill. That is why for a little while, you do not have all of the abilities you have by the end of the game. It funny because since I played AC II first, the references to Altair really made me think he was awesome, which is what made me play the first. I had heard that the first game was not that great, so I never really intended to check it out until I knew the codex pages were written by Altair, then there was his statue, tomb, and armor. Then after I played the first game, I realized I did not really like him and thought there was nothing special about the character. I will have to play the first one again at some point and see if I like him any better.