Is it acceptable to write: A person can develop their talent. or a person can develop their uniqueness? In this case the person is used as a general term, not a specific person.
The question here is not really whether or not person can be used in the plural, but whether or not they (and its relatives) can be used in the singular. "Singular they" has a long and strong tradition in English, going back to a time long before the idea of gender neutrality became a concern. Feel free to use it in any formal or informal writing where it is not prohibited by a style guide (or an editor with strong, though misguided, opinions). Yes, some self-appointed grammar mavens may crow and grumble. Let them; they think it's fun, and you'd be doing them a favour.
This is not a plural as much as a nonspecific singular. "Their" is being used as a singular they and as such does not imply plural.
This is an unusual case as you are referring to a single, but unknown person as an example of people in general, I would probably say that this is correct but there are better methods of phrasing this for clarity, such as:
One can develop one's talent
As mentioned by @Ben in the comments. This is a more specific way of making it obvious you are using an nonspecific singular.