This seems to outline two different issues. The first is the ownership issue -- who owns the fiction, is it entirely the GM or do the players also have some ownership? Let's call this Fiction Control. The most common approach here is heavily weighted to the GM having full and complete ownership of all fiction outside of the character, including external elements of any character background. You see this quite often in complaints about the PCs pushing new things not planned on into the fiction of the world and in statements about veto authority over background fiction introduction. On the other end of this is being perfectly okay with players introducing new elements into the game and having the authority to do so. This approach is very uncommon in the D&D community, though, although it does exist.
The second issue is what kind of game is being played? Let's call this Plot/No Plot. Is it a plot driven game, where the play is expected to be about a specific plotline or element (adventure path play being a prime example of this? Or, is the game driven by what the characters do, and the next moment emerges from this and isn't tied to a plotline? The latter is becoming less common in D&D, but is characterized by the dungeon crawl, where the goal is to solve the dungeon but not adhere to any particular path or plot while doing so.
These combine in interesting ways. If he'll forgive me,
@iserith seems to be on the permissive end of fiction control for backstories, and then a non-plot driven game in the second issue, so the first doesn't really impact the second unless the players make it so -- the GM doesn't care. Games that are heavy GM fiction control and heavy plot are very adverse to players introducing new elements into background. To pick on him,
@iserith just above posted a good example of a game where the player have significant fiction control to introduce new material in their backstory paired with a strong plot play. So, these can mix and match, but most answers to this fall along these two main axis.
I used to be strong GM fiction control, strong plot, but now am more player fiction control and weak plot to no plot. Well, except that I am running an AP right now, so my most recent 5e game is strong plot. My last few haven't been.