Etymologically, the word carries a sense of “put someone under someone else’s control.” Meaning “to give up absolutely” is from late 14c. Related: Abandoned; abandoning.
“a letting loose, surrender to natural impulses,” 1822, from a sense in French abandon (see abandon (v.). Borrowed earlier (c.1400) from French in a sense “(someone’s) control;” and cf. Middle English adverbial phrase at abandon, i.e. “recklessly,” attested from late 14c.