My guess is that not many CIA operatives come renting scooters on Mykonos.
Besides Bourne told the CIA she was dead so they most likely wasn't looking for her at all. The bag only becomes a liability when Bourne shows up again at the end.
Another possible scenario is she never expected to see Bourne again so she might have kept that bag as a memory of him. Love makes people do stupid beep sometimes.

Visual symbolism. Everything Marie owned was in her red car. It was the only thing she took care of herself. (The "splashy tires" discussion right before the epic chase scene). After the chase scene it was sort of "their" car. Everything Bourne owned was in the red bag. She had to transfer the contents of her life when she lost her car and in some sense her identity. He eventually gave her his red bag just as she had given him her car. They couldn't keep the car but she could hide their bag.

Yup, and it's the only thing he keeps throughout the film, but why? It would be the first thing I'd ditch

Holy crap I never noticed that before, and I've seen this movie a hundred times. I just watched the clip on YouTube. You're right, she'd be insane to keep that red bag hanging in her shop like that. It's a dead giveaway.

noo it was a motif throughout the film, she even had flowers in it at the end. Just seemed odd to keep it given that it was a clear flag to anyone pursuing him

And replace it with what? How?
It was the waste bag he used to take all the stuff from the lockbox in the Zurich bank. He was improvising.