Never saw one without the other
I saw a take the other day that this is overplayed in the fandom (which is impressive considering the fandom tends to go overboard to emphasise Sirius and Remus' friendship, but okay). I would argue that it's usually underplayed, but what is overplayed (also by me) is how much this would (or as it happens, wouldn't) stand out.
In Harry Potter, there are quite a few people you never see without the other:
Crabbe and Goyle (and Draco)
Harry and Ron (and Hermione)
Fred and George (and Lee)
It is normal for teenagers to hang around with the same people to the point of being inseparable, especially at a boarding school. Nobody goes around describing Ron and Harry as never seeing one without the other, nor Dean or Seamus etc but it's clear to us readers that these groups take the same subjects and hang out together when they have the chance. The only ones that are noted as being hard to spot without the other are Fred and George (and Lee, as it happens).
Then there's the fact that MWPP are described like this on Pottermore:
Soon, these four became inseparable.
AND YET, despite the fact that the four are pretty much inseparable, it is James and Sirius who are described this way by their teachers, the Minister of Magic and Rosmerta:
“You say you remember him at Hogwarts, Rosmerta,” murmured Professor McGonagall. “Do you remember who his best friend was?”
“Naturally,” said Madam Rosmerta, with a small laugh.
“Never saw one without the other, did you? The number of times I had them in here — ooh, they used to make me laugh. Quite the double act, Sirius Black and James Potter!”
(...)
“Precisely,” said Professor McGonagall. “Black and Potter. Ringleaders of their little gang. Both very bright, of course — exceptionally bright, in fact — but I don’t think we’ve ever had such a pair of troublemakers — ”
“I dunno,” chuckled Hagrid. “Fred and George Weasley could give ’em a run fer their money.”
“You’d have thought Black and Potter were brothers!” chimed in Professor Flitwick. “Inseparable!”
“Of course they were,” said Fudge.
They are literally compared to Fred and George, who you really never see without the other except two occasions, both of which end badly (and we don't need to think about the second one).
Also, there isn't one memory in which we see James, except the very last one when Voldemort murders him and Lily, in which he's not with Sirius. Even the prequel features JamesANDSirius, and not the Marauders as a group. When Lily writes to Sirius, she can't help but mention how much James would appreciate seeing him. These aren't things that are there by chance. They are there to paint a picture of a duo (trio in the end), who are used to being together, always. Only to have Sirius lose James (and Lily).
Things I therefore don't find weird:
Sirius playing Quidditch. The man seems to greatly care for the sport to the point of watching it when he's on the run and buying Harry his first broom and his Firebolt. I am not saying he have to play Quidditch; personally, I go back and forth on this all the time, and if he played he certainly wasn't in James' league given we keep hearing about Snape's jealous of James' Quidditch talents, but I definitely don't consider Sirius playing Quidditch canon breaking.
Sirius and James taking the same electives at school. On this one you cannot convince me that they took different electives to one another. Most of the students seem to take the same classes as their friends, so why wouldn't the two people whose teachers note are inseparable do so?! Also considering that Bill, Percy and Crouch Jr took twelve OWLS (possibly thanks to JKR's poor maths, because you cannot convince me all of them were handed timeturners, and which is why a few speculate that it was easier to take more subjects before), there is even a case to say that "exceptionally bright" Sirius and James would have taken all twelve subjects (but NO timeturners, no one in their right mind would give Sirius and James this).
What I can get onboard with is that making this a big deal in fanfiction is, perhaps, a bit much. Teenagers stick together. James and Sirius stuck together. Not because they couldn't function without one another (as much as it pains me to admit this), but because they enjoyed the same things and liked each other's company. No one would really think that hard about it.
For this to work, though, they do have to be together pretty much all of the time, at least until James starts going out with Lily. Even then, while I weirdly believe that James would go on dates without Sirius (gasp!), we do see that Lily and Sirius do become close, suggesting that oftentimes the three of them (or maybe also Peter) hung out.
Which is why I suspect that the real cause isn't that it's overused, but underused, and that the way it's being handled by the fandom is to focus on telling the readers that they were inseparable, rather than showing it!
Also, as the whole angst point here is that Sirius loses James at 21, I will never stop making "never saw one without the other" a point that people think or say or reflect over, to make it obvious to the readers that Sirius' world shattered only a few years after school. Deal with it!