I can only speak for myself, but when I barrack for the Melbourne Demons, I am not supporting Collins St. or the MCC or Federation Square or anything like that. Nor, I guess, do most Collingwood, Geelong or Hawthorn supporters. I think we all barrack for the feeling we get after a come-from-behind win when we're little, and the resulting visit to McDonald's, or the brand new scarf our grandpa bought us one Friday night, or the feeling that my name's Jimmy, my grandpa's name's Jim and there was a guy named Jimmy Stines out there single-handedly kicking butt for the nine-hundredth game in a row.
I suppose I'd feel all of those things no matter what club I barracked for. So in answer to your question, "Melbourne" just happens to be the name of the team with which I associate a large part of my better childhood memories. So I support the team for the same reasons that I choose to remember the good times - because my life would be that much duller if I didn't.
And because you'd be crazy to support anyone else. I mean, as we all know, every heard beats true for the Red and the Blue.