To enter a thing on an official list. Also the particular pitch and manner a voice speaks in.
Late Latin registrum/regestum, from Latin regerere (to carry back, to record): re- (back) + gerere (to carry, to conduct); to register is to carry an entry back into an official record, and the same root's carrying gives the word its other sense, the pitch a voice is carried in.
To enter a thing on an official list, or the particular pitch a voice speaks in. Both senses share the root's carrying: something is conducted into a fixed place, whether that place is a ledger or a range of sound.