So they look incredibly similar to another missing person and are mistakingly identified as the other person and the hospital later finds out they’ve made a mistake after they’re back with the other person’s loved ones / family and they don’t notice the difference
What would happen and has this happened before
AFAIK complete amnesia is rarely permanent, so it would probably fix itself most of the time if there’s no serious compounding issues.
Either way, the likelyhood of two missing people in the same general area looking so similar that their family can’t tell them apart visually seems extremely low.