Its less about rights and more the principle of the matter.
Rae made a choice for herself, probably one of the first real choices she's made in her life, one that was influenced or decided or allowed by someone else. It was her decision, pawn an item that doesn't mean that much to her for something that is solely hers and hers alone. Its clear that Rae doesn't own many things, look at the joy she expresses when the MC manages to bring most of her bedroom to the house. Being sent to other worlds, often years at a time, she probably never had the chance to simply be a young woman, not a servant, not an object of desire, not a tool, just a young woman.
Sure, the MC buying the armband back and giving it to her can be seen as sweet, after all, it was no longer hers. But from Rae's perspective, the MC pretty much undermined the exercise and treated her like so many others have, as an object to be fawned over and given pretty things.