I think you are correct. It’s hard to know exactly what the writer means with certainty. Their identity would mean that they share the SAME identity. Their identities would suggest that each person has a separate identity. OR, the writer may have used their (incorrectly) rather than writing « where his or her identity can only … ». That would be correct, but is a bit cumbersome so most people avoid that construction. Please note - English speakers and writers make lots of mistakes when it comes to possessive pronouns. There is also a growing trend to use the plural possessive pronoun (their) instead of the singular possessive pronouns (his/her) because the plural pronoun is not gendered whereas the singular are. And it also allows you to avoid « his or her ». In general: Their is the plural third-person possessive pronoun. It can be followed by a singular or plural noun. Example : their house or their houses. In both cases, the item(s) belongs to two or more people. His, her and its are the singular third-person possessive pronouns. Its is not used for a person. Like the plural, these can be followed by either a singular or plural noun. Example: his car, her cars. In both cases, the item(s) belong to only one person. Bob and Sue own the house. It is their house. Bob and Sue own two houses. They are their houses. Bob owns a car. It is his car. Sue owns three cars. They are her cars.

I think it’s because the ‘their’ is covering both the man and the woman. If you change their to ‘his or her’ it probably makes sense.

"Identities" would be fine too and that would emphasis that there are two of these mystery characters that are being addressed. "Identity" implies there is one identity as if the two mysterious characters are really just one identity or one pair, if you like. Someone who knows more about grammar than I do may have a better answer, but perhaps that is a start?