It seems like the filter would pick up the odors from the refrigerator, and then pass them on to the water. Or no?
While I don’t know about Brita filters specifically, you can buy activated charcoal for absorbing fridge odors, so I think it’s certainly possible. However, I don’t know how well the filter remains wetted, and my hunch is that wet charcoal wouldn’t absorb odors as well as dry because I think many odor molecules are hydrophobic.