Maybe not completely different, depending on how close we look. It's like a fractal -- the closer we zoom, the more distinction we see.
Can we consider gaming a hobby? Sure. Let's look closer.
Can we consider tabletop games a hobby, separate from video games or (I don't know the word) games like football and basketball? Sure. Let's look closer.
Can we consider TTRPGs a hobby, separate from wargames, board games, CCGs and gambling? Sure. Let's look closer.
Can we consider OSR a hobby, separate from New School and Mid School? Sure. Let's...
Yeah, let's stop. I don't really want to type more of the same naughty word.
I didn't say they are mutually exclusive. I did say that they are very different.
If you are running 5E as an impartial judge, you're engaging in a Old School gaming. If you are running B/X with the focus on actively creating a story (and not just getting one from series of rolls), by invoking genre tropes and taking into account story structure (three act structure, five act structure, whatever else you've picked), then, well, you ain't doing it in Old School way.
Yeah, and hip-hop is hip-hop. But no one would argue that cloud rap is very different from old skool. Though, I'd definitely say that "old" and "new" are very stupid terms -- a band that was calling themselves "New School of Russian hip-hop" is now considered aggressively old-school
And, well, I don't know naughty word about collecting hot wheels or matchboxes... Oh. Damn. Ok. I thought you were comparing matchboxes (like, boxes of matches? I guess someone collects them) to hot wheels. Sorry, disregard that. The only brand of toy cars I know is hot wheels.
Though, continuing with the things I know, within firearms collecting there are very separate categories. Collecting Wild West guns is very different from collecting WWII guns, which is very different from collecting AKs (no one asked, but this is what I do).