It's a bit more varied, even in the Indo-European family. What does tend to happen is that the words for handedness get positive (right) or negative (left) associations in idioms, but additional meanings are not universal. In French, "droit" additionally means right (as human right), but not "correct" (yes it does have a bunch of adjacent meanings). In German, "recht" gets to mean "law" or "justice", shared by some Slavic languages ("pravo") -- but not all of them, which have the word "desno", without any association with rights/justice/correctness. The Latin "dexter" gave us "dexterity" and "dexterous", but also nothing alluding to justice. Et cetera.
As an aside, "left" originating from "left over" sounds like folk ethymology to me. Dictionaries point to "weak" as the original meaning.