I was thinking face-masks.
People usually focus in on the face when identifying somebody definitively, which means a face mask will be front-and-center when it comes to identifying people. Additionally, if people are very against being mistaken for other people, than focusing in on the little identifiers (clothes) is probably less effective than addressing big identifiers like having the same face.
How would it work? I would guess cloth face masks, since they will be more comfortable to wear long term and light to carry. It would also be pretty easy to personalize, too, between cloth choices, embroidery, and other decorations. They don't have to be full face masks, either, choices among partial masks can be an additional layer of identification, with one person preferring a full face mask, another just an eye mask, and a third a mask over one side of their face. And one pocket or pouch full could carry a lot of masks, for a person to change into.
Culturally, the mask would then be your "face" in a way skin-and-bone would find problematic, when bodies may be shared among so many people. It would be no more abnormal to them than us wearing other kinds of clothing (or identifying things about people from such). There could be a set meaning to going bare-faced, like signalling changing personas, or it could be a privacy taboo, where people do not bare their faces without a very good reason and perhaps people would keep a blank mask to have something to cover their face with if something happened to their personalized one.
If, for whatever reason, it is important to identify people form behind or without seeing their face - it may be useful to pair with scarves or shawls. I mean the light, thin ones large enough to drape (easier to carry multiple, then), possibly reversible to maximize the number of variations for less cloth carried. Color plus pattern plus wearing habit (tied shoulder-to-hip, draped around neck, wrapped around shoulders, as a belt) can again individualize a person pretty distinctly without seeing the face and without having to make elaborate changes. I think it would still be best paired with masks, since it's easier to notice differences like sash or scarf when not using the same face for identification.