An American perspective:
I reschedule my plans to watch my team on TV.
I buy $100 jerseys and spend $100's of dollars other stuff with the team name on it.
I pray for the team to do good (no litterally in the middle of the game I'll get down and say 'God please help this field goal go in').
Everytime the team looses I feel like a small part of me has died.
Everytime it wins I feel like that little part has come back to life.
I pay for $50-80 tickets to go to the games.
I spend the entire offseason on forums and websites discussing what the team should and shouldn't do.
So ya, when I say "us" or "we" or anything else like that I do feel like I belong to the team. I feel like, as a whole the fans, of any team are an extension of that team. They are considered the "6th man" or "12th man" (depending on the sport of course). They have given themselves to the team. They like them no matter what. I happen to be a die-hard fan of a team that hasn't had a winning season in 10 years or more and who regularly finish at the bottom of the standings but they are "my team". I feel a sense of pride when I wear their jersey or fly their flag in front of my house or drink out of my team colored coffee cup. Just because I don't actually play, I'm still representing the team. And I'm sure I have more loyalty to the team than 80% of the players who will leave at the drop of a hat if the money is right.
A great example from the NFL is the Seattle Seahawks two years ago. They crowd at the Seahawks stadium is called "The 12th Man" because they are so loud. They were making an improbably run towards the Super Bowl (they acctually had a losing record but made the playoffs). They were trying to run out the clock and suddenly their running back made a huge play (he broke like 6 or 7 tackles and made about a 60 yard run - Marshawn Lynch) that ended up being the "dagger" that sealed the game. The stadium of course erupted - litterally. There was activity measured at an Earthquake detection center and the epicenter was litterally the stadium. Now you tell me these fans don't have a right to say "us" when they talk about the Seahawks.
Here's the vid:
Here's a news article: Marshawn Lynch 'Beastly' Touchdown Run Causes Minor Earthquake in Seattle