Proper comes from Latin proprius meaning 'one's own, particular'. The shift from 'belonging to oneself' to 'correct' happened because what is truly yours is genuine, and genuine became appropriate.
Truly what something is said or regarded to be; genuine; appropriate to the circumstances; correct in behaviour.
From Old French propre meaning 'own, personal, individual', from Latin proprius meaning 'one's own, particular, special'. The origin of proprius is debated: it may come from prō prīvō ('for the individual') or from an older Italic form. The core sense is 'belonging to oneself' — something proper to you is genuinely yours. The shift from 'own' to 'correct' happened because
In French, propre still means both 'own' and 'clean' — cleanliness is what is proper to a well-kept person. In Middle English, proper also meant 'handsome' — Chaucer called a man 'a proper knight', meaning good-looking. Property comes from the same root: your property is what is properly (genuinely) yours.