1530s, "rights and powers of a king, royal privilege," from Latin regalia "royal things," noun use of neuter plural of regalis from rex (genitive regis) "king" (from PIE root *reg- "move in a straight line," with derivatives meaning "to direct in a straight line," thus "to lead, rule").
The meaning "decorations or insignia of an order" is recorded from 1670s, probably via the sense of "the emblems or insignia of royalty," e.g. the crown, scepter, etc. (1620s).
In paraphernalia, Mammalia, regalia, etc. it represents Latin or Greek -a (see -a (2)), plural suffix of nouns in -ium (Latin) or -ion (Greek), with formative or euphonic -i-.