via notion of "tramping the feet." Noun meaning "act of marching" is from 1580s. The musical sense first attested 1570s, from notion of "rhythmic drumbeat" for marching. Marching band is attested from 1955.
(obs.) "boundary," late 13c. (in ref. to the borderlands beside Wales, rendering O.E. Mercia), from O.Fr. marche "boundary, frontier," from Frank. *marka (cf. O.H.G. marchon "to mark out, delimit," Ger. Mark "boundary;" see mark
c.1200, from Anglo-Fr. marche, from O.Fr. marz, from L. Martius (mensis) "(month) of Mars," from Mars (gen. Martis). Replaced O.E. hreðmonaþ, of uncertain meaning, perhaps from hræd "quick, nimble, ready, active, alert, prompt." For March hare, proverbial type of madness, see mad.