It's not really a question of who they belong to. They belong to the airline, which keeps track of them on your behalf. You don't "own" anything.

The real question here, though, is what is the airline's responsibility in managing your account. They owe you a reasonable level of care in this regard. Based on your post, they haven't shown it. If you have any evidence to back up your statement that it was the check-in staff's error, I'd pursue it further with the airline. They can't duck out of responsibility for their own staff's errors by simply refusing to fix them, especially when fixing them is still possible.