According to The Online Etymology Dictionary:
early 14c., "piece of armor for the arms," also "thong, strap for fastening," from Old French brace, braz "arms," also "length measured
by two arms" (12c., Modern French bras "arm, power;" brasse "fathom,
armful, breaststroke"), from Latin bracchia, plural of bracchium "an
arm, a forearm," from Greek brakhion "an arm" (see brachio-). Applied
to various devices for fastening and tightening on notion of clasping
arms. Of dogs, "a couple, a pair" from c.1400.
mid-14c., "to seize, grasp," also "wrap, enshroud; tie up, fetter," from Old French bracier "to embrace," from brace (see brace
(n.)). Meaning "to render firm or steady by tensing" is mid-15c.,
earlier in figurative sense "strengthen or comfort" (someone), early
15c., with later extension to tonics, etc. that "brace" the nerves
(compare bracer "stiff drink"). Related: Braced; bracing.