Good topic.

I think that first of all, you've got fans of basketball who love the sport, plain and simple. They're passionate about watching the game (and probably playing it too, their own skill level not withstanding) and they keep up with the news, scores and results from professional and amateur leagues alike to varying degrees. Not everyone is a fountain of knowledge but they'll at least know the basics of basketball. For some, basketball may be one of many sports they enjoy but the more hardcore/enthusiastic basketball fans may only watch basketball or at least place it atop the list of sports they like. A real basketball fan simply loves the game of basketball; loves watching it, loves reading about it, loves talking about it.
Then you've got people who like a particular team and/or specific players on top of that. The reasons for being a devotee of a particular team or player(s) varies from geographical location to the first game you watched to yes, media hype and existing popularity (or alternatively, latching on to the underdog). Such fans can at their core simply be huge basketball fans as well but they stick by a team through thick or thin. Fan loyalty has its limits though and certain actions by a player or team could ultimately lose the allegiance of even the most diehard fan. A team moving cities, some sort of huge public scandal or simple player movement and bad decision making would be examples of things that test the patience - and loyalty - of a diehard fan of a team or player.
After that you've got various subgroups with plenty of overlap such as fans who do love basketball but can only really take an interest in watching it when their favourite team is playing, are homers for their team, are dedicated to their team but can enjoy watching other teams and players play, etc. I think it's fair to call such fans "real", though they have different and conflicting ideals and approaches to their fanaticism.
As far as people being fans of players or teams, I think there's examples of both within the fanbase but given the amount of times you see fans turn on a player when he's traded, downplay his contribution to the team and boo him upon his return - regardless of what the circumstances may actually be - I think the allegiance to teams is still very strong and while some fans may still follow a player and want to see him succeed when he's wearing a different uniform, it takes drastic circumstances to shift that allegiance away from the team they've adopted as their own.
As for the opposite camp, which I suppose would be the "fake" fans, those would be the bandwagoners who switch allegiance frequently while claiming to be diehard fans of whatever team they like that year/month/week/day/minute, though they're probably still true fans of the sport itself. Then you've got the fair weather fans who will support their team as much as the diehards...so long as they're winning, otherwise they're noticeably quiet. Then you've got the two groups that appear in pretty much every fandom: the people who like the sport or a particular team/player simply because it's trendy and stop being fans the minute it ends, along with their polar opposites the people who only like something while it's rare/against the trend to do so and then stop as soon as it gets popular (think of people who like a certain band or artist while they're a relatively unknown then claim they sold out/suck now that everyone likes them) because they're really fans of being fans of something that's different, more than they're ever fans of whatever it is they claimed to be fans of...if you can follow that.
And somewhere in between you've got the casual fans who have an appreciation for the sport and possibly a team or player(s) as well but never claim to be anything more. They often get lumped in with the bandwagoners though as they usually swell the ranks of the fanbase during the good times and to be fair with some casual fans there probably is a bit of overlap with the bandwagoners.
So, what makes a real basketball fan? Passion and keen interest in the sport that is practically unwavering, a devotion to teams and players being optional. A real fan of any given player or team similarly shows practically unwavering support to their chosen team/player(s) through the good times and bad, not jumping off the bandwagon no matter how they came to be on it in the first place unless a certain line is crossed. I'd also make the distinction that supporting a team or player does not necessarily constitute agreeing with every move/decision they make or believing they are the best when they clearly aren't, but supporting and cheering for them all the same.