1570s, bragget, "architectural support, supporting piece projecting from a vertical surface," probably from Middle French braguette, originally "codpiece armor" (16c.), from a fancied resemblance of architectural supports to that article of attire (Spanish cognate bragueta meant both "codpiece" and "projecting moulding in architecture"), diminutive of brague "knee pants," ultimately from Gaulish *braca "pants," itself perhaps from Germanic (compare Old English broc "garment for the legs and trunk;" see breeches).
The architectural meaning also might be directly from the "breeches" sense, on the notion of two limbs or of appliances used in pairs. The senses have been affected by Latin brachium "arm." The typographical bracket ("marks used to enclose a note, reference, explanation, etc.;" each was formerly a crotchet) is first recorded 1750, so called for its resemblance to double supports in carpentry (a sense attested from 1610s). Meaning "position of being enclosed or bracketed with others" as of equal standing is from 1880.