What is placed around a thing to hold or present it. Wrapping is not the thing and changes how it arrives.
Middle English wrappen (of uncertain further origin, attested from c. 1300) plus the Old English suffix -ing, which turns a verb into the noun of its result; wrappings are the accumulated material a wrapping act leaves behind — the covering considered apart from what it covers.
The covering that gets set aside once whatever it held or presented has actually been reached — never mistaken for the gift itself, however carefully it was chosen. What arrives changes with the wrapping; what is inside does not.