I think you mean (at least I hope you mean) in your roleplaying games. Because apparently a lot of people here do allow players to contribute to the narrative beyond their own character. Do you think they/we are all playing wrong?
This thread is asking for opinions - Would
I allow it? No, I would never allow such a thing. Why not? Because it's not role-playing, and I play role-playing games for the purpose of role-playing.
A lot of people would allow it. That's fine, for them. To me, it says that role-playing is not their priority in a role-playing game, and they'd rather tell a story
about their characters than make decisions
as their characters. I probably wouldn't have fun at those tables, because (as previously mentioned) I'm there to role-play. I'm willing to commit five hours per week to pretending to be a magical elf, but I'm not willing to spend five hours per week on collectively improvise a story
about a magical elf and their friends.
(It's debatable whether a game even qualifies as an RPG if you're just telling stories rather than actually playing roles, but this isn't the place for that debate.)
Would you be ok with that? Or is the player overstepping by just making up all of that...the scarf, it's meaning, etc.
It's over-stepping, because the player is inventing details about the scarf and so on, which are not things that the character has any control over. The inherent meaning of that scarf was already set in stone before the character started talking, and nothing that they say would be able to change that meaning.
It's also
under-stepping, though, because the player isn't making the decisions that their character
would have control over. The character
would have absolute control over their approach to the situation, and what they want to say, but the player seems to have relegated that decision to the die roll.