lol no. It's an emotional subplot. "WINNER" means he basically just gets a free Sea-Salt Ice Cream (it's something from the KH2 novels I believe). The point was, he brought it up a lot in his diaries that he was going to ask Axel what it was about, but he never did. It's just something to add to their friendship and what not.
Pretty much this. In a short story book published while Days was being finished and written by the scenario writer for Days, there were several mentions of a white envelope that Axel looked at sadly and carried around while he was being ordered to kill Roxas. Playing Days reveals this to be the winner stick.
Roxas had been saving this stick all along to share with Axel. At first he was saving it up to have just with Axel. Then he started to care about Xion too, and he wanted to wait until he had another one before he used it. It was kind of a symbol of the difficult balance in a group of three. In the end, though, no one got free ice cream. Roxas left, and he left it with Axel. I think it then to Axel it became a symbol of the loose threads between him and Roxas, like, things they promised to do and never got to in the end.
I'm not quite sure what Roxas meant by leaving it, though.
+here, you take it, i'm not going to need it cause i'm out of here
+have this, i still kind of care about you despite everything and i wanted to give you free ice cream
+have this dickface, it's your 'reward' for being a backstabbing liar
+have this to remember me by
etc etc etc
lol