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.
The tendency to associate the right with positive and the left with negative is pretty much ubiquitous in Western civilization up until fairly recently.
I have a fairly exotic viewpoint on this. The left/right asymmetry (for the lack of a better term) is a small part of a larger asymmetry that's difficult to explain (at least to my satisfaction): up is better than down, light is better than dark, positive is better than negative, higher human faculties are better than lower desires, right is better than left. In my mind, none of these should inherently have a good/bad association. Is a valley worse than a hill? Is a positive electric charge better than a negative one? But, for some reason we have this cultural, linguistic baggage where what seem like highly abstract objects are rooted in a very basic world of good and bad.
I find this fascinating.
I would go as far as to add the male/female asymmetry to this: seemingly senseless, yet historically present. I'd say the rabbit hole truly starts when you get into Western esotherics and see how they associate right, up, light, male, order, knowledge as if they're different expressions of the same thing, and analogously for left, down, dark, etc.
Makes sense; up gives you a better vantage point, so its easier to see danger. Not to mention that it gives you a fighting advantage; your attacker would have to run uphill to get to you, tiring them out.
> light is better than dark
Humans rely heavily on our sight, and we tend to operate in the daytime. If you were a nocturnal animal, I'm sure you would consider dark to be better than light.
> positive is better than negative
Getting something is better than giving it away.
> I would go as far as to add the male/female asymmetry to this: seemingly senseless, yet historically present
What do you mean by this?
In clothing you have the right side and the reverse side while here it's oikea ja nurja. Again similarities but nurja has this twisted and dark connotation. Or upside down = nurinperin.
Edit: or maybe indo-european and semitic to be more broad.