You're asking logical questions about something, "fandom", where "fan" is short for "fanatic".
To me, you're a fan because of emotion, not logic. I grew up watching the Bears. That's the team I rooted for, that's the team my parents rooted for, that's the team that the city of Chicago rooted for. It wasn't logical. It was just "our" team. In reality, the Bears were not really anything to me. I've since lost that fandom and I'm now a "homeless" NFL fan... I don't have a team.
I started and still root for Purdue. I didn't grow up rooting for Purdue. Nobody in my family went to Purdue, nor did my family even care about college sports as both my parents went to colleges without robust athletic programs, and Chicago was a very pro sports dominated city. But I went to Purdue. I became a Boilermaker. I cheered in the stands with my closest college friends and lived Purdue wins and losses because it's my school. Is it logical? No. I wasn't an athlete. Purdue's performance on the football field or basketball court didn't really have any relationship to the quality of instruction I had in electrical engineering. In short, they were decoupled. And yet I still care and I'm still a fan.
The logical thing is to be fans of the best teams in each sport, and to throw teams aside if they are usurped by other teams. Logically, that's the route to the most fan happiness. But to me that's a soulless, calculated fandom. It's manufactured.
I could never do that, because I know that if a team doesn't give me a negative emotional reaction when they lose, then I don't care about them enough to truly enjoy or get a positive emotional reaction from a win.
Fandom isn't about winning championships. It's about an identity or experience that connects you emotionally with other people. And those connections are built much more strongly sticking through down years together as it is enjoying the wins together.
I would suggest we could modify the definition of "fan." It doesn't have to be tied to a particular team, although at one time I would've disputed that, and granted, it's a different kind of fandom. We could use the concept of "enthusiast" as an additional definition of "fan." So that, while I'm a homeless NFL fan, like you, I nevertheless consider myself a fan, in that I like to watch the games, I appreciate the sport, and I miss it when it's not there.
For most of my life that wasn't true. For me, NFL < rat's ass. I didn't watch it and didn't care. Now I do watch it, and enjoy it. Clearly, something has shifted in the game's relationship to me, and it seems incorrect to deny some type of fandom.
For the most part, it doesn't matter who's playing, I just like to watch the games. So while I don't "have" a team (and I'm not sure how I could force myself to go about that process), I'm still a fan of the game, if not a particular team. And there are tons of players I enjoy watching. And then there's the element of how I like to keep an eye on former LSU players and genuinely root for them to have a good game, complete with some level of sadness if they don't. But that's spread out across nearly every team in the league.
And by your definition, I still think it can be viewed as an identity and shared experience with others.....it's fun to come on here and talk about something interesting that happened in a game with other fans, even if the ending didn't elate me or gut-punch me like it surely did some others. Though it obviously lacks the element of stronger connections due to weathering tough years and the same sort of shared joy from a great year.
I have as much of an emotional tie to the Cowboys as is possible for my background (unless I figure out a way to manufacture a fandom more in line with my LSU fandom). It's rooted in being married to a fan and likely having some level of empathy for her emotional interests, and then watching all the games with her for the past several years that we've been married (familiarity doesn't just breed contempt, it can also breed affinity....or maybe it's Stockholm Syndrome, not sure). But it obviously lacks elements that, say, her Cowboys fandom has. But that doesn't mean I don't love watching them play. Because I do. I might love watching many teams play, but to me that just reiterates that I'm a fan of the sport and not a team, but a fan nonetheless.