The behavior you’re describing could be associated with a few different terms, but a person who is particularly concerned about maintaining a neat and polished appearance might be referred to as "fastidious" or "meticulous." If they are also somewhat assertive in setting boundaries with others, you might describe them as "discerning" or "particular."
If you're looking for a more casual or conversational term, you might simply refer to them as "high-maintenance," though this term can sometimes carry a negative connotation. Another possibility is to say they are "prissy" or "fussy" in how they prefer to present themselves and interact with others.