As it should be, by the standard that applies. Also belonging to one particular thing — a proper name names one.
Latin proprius (one's own, particular, special), related to prope (near); a thing is proper first because it belongs distinctly to itself, and only from that belonging does the sense of correctness follow.
As it should be, by the standard that applies, and also belonging distinctly to one particular thing. Both senses share the same root claim of ownership: a proper name and a proper way of doing something both mark out what is one thing's own.