It is sometimes used that way, but is (I think more) often used in a much more analytically useful sense: as a means of indexing passive/automatic advantages that some people benefit from, and as a lens into the ways that advantage might be extended to others.
> This seems to imply that those with privilege think their privilege is a right.
That doesn't follow logically (if group A is bothered that the advantages of group B are not considered to be group A's right, it does not follow that group B considers those advantages to be group B's right), or textually (GP referred to "the world" as the thing not considering privilege a right for some people, which I interpreted as a reference to the context/emergent behavior/power structures that confer advantages; not specific people).
> In that sense it feels perversly exploitative and priveleged to make so much more money ...
It seems to be more useful for shaming people.
> ... that the world does not consider this to be their right.
This seems to imply that those with privilege think their privilege is a right.