Old English anfilt "anvil," a Proto-Germanic compound (cognates: Middle Dutch anvilt, Old High German anafalz, Dutch aanbeeld, Danish ambolt "anvil"), apparently representing *ana- "on" (see on (prep.)) + *filtan "to hit" (from PIE root *pel- (5) "to thrust, strike, drive").
The ear bone (incus) is so called from 1680s. Anvil Chorus is based on the "Gypsy Song" that opens Act II of Giuseppe Verdi's opera "Il Trovatore," first performed in Teatro Apollo, Rome, Jan. 19, 1853.