The thing is, I learned way back in grade school that “give” was a verb, the act of taking something and passing it to another person, and that “gift” was a noun, the object of that action.
To me, using “gift” as a verb is the same annoying mis-usage as saying “I bequest my home and fortune to my loving wife Betty” or “I donation $500 to the Red Cross every year” rather than the verb forms bequeath or donate.
Oddly “re-gift” (and other made up words like google and tweet [like on Twitter]) don’t bother me at all, probably because they were made up out of whole cloth and there weren’t words used to describe taking those actions before.
The first time I heard “gifting” as a verb was in relation to the “gifting club” pyramid schemes that were big a decade or so ago. That’s another possibility for the word’s entry into wider usage.
I first saw it in writing in text written by someone in India. I was proofreading the text and changed it. The writer changed it back and (politely) asked me WTF, as it is apparently perfectly normal Indian English and had been for some time. This was in 2001 or 2002 or so.
The earliest I remember gift and gifting used as verbs was around 1989. A department store chain was using it in its Christmas advertising in the newspaper and on TV. I remember people mocking the usage around that time but never noticed anyone using it for real until the past few years.