As Edwin Ashworth says, the tendency among current grammarians is to call away in this sort of context ( give away, put away, send away) a ‘particle’.

What is the moral significance of the new catchall trashcan term?

I suggest that these ‘particles’ are in fact intransitive prepositions—prepositions who stand alone, without obliques, and thus constitute in themselves preposition phrases.

Whereas away is a preposition phrase in ancient Greek, as in similar terms like aboard, adrift, aground, ahead, aloft, all is worn down form of an = ModE “on”. Many modern intransitive prepositions have preserved their prepositions more discernibly ( downhill, upstairs, overhead, underfoot ), others have even collapsed their prepositions into their obliques ( home, east ).

This is I think clearly justifiable with verbs of motion and caused motion, where away plays the same syntactic role as ordinary preposition phrases. (place her) Go to London/away, put it on the table/away. The occurrence with give is only a little more opaque; Goldberg, Constructions, 1995, notes the very close affinity between ditransitive constructions, which usually allow the Recipient to be realized as PP rather than an IO (“dative alternation”), and caused-motion constructions. What are the most common conjugations for giving (give, give back, give off, give over, give over, give up ). Note that there are transitive uses for these prepositions, of which many are employed both with and without direct objects.

Why is always-intransitive preposition a locative complement of give, as in a totally conventional choice of direction or (indeterminate) goal of the gift.