'Remove' and 'remote' are from the same Latin verb — 'remote' is literally the past participle of 'remove.'
To take away or off from a position occupied; to eliminate or get rid of; a degree of distance or separation.
From Old French 'removoir' (to move away, to take away), from Latin 'removere' (to move back, to move away, to set aside, to withdraw), composed of the prefix 're-' (back, away from — here carrying its original spatial sense, not repetition) and 'movere' (to move, to set in motion). 'Movere' traces to PIE *mew- (to push away, to move), which also produced Latin 'momentum' (movement, impulse), 'mobile' (capable of movement), and 'emotion' (from 'e-movere,' to move outward — a feeling that moves out of one). The English word arrived via Anglo-French in the 14th century with senses
The noun 'remove' meaning a degree of distance ('at one remove,' 'at several removes') preserves the original spatial meaning of Latin 'removēre' — a moving-back, a distance. This sense is now literary, but it was once the primary noun meaning. The word 'remote' (from Latin 'remōtus,' past participle of 'removēre') is literally 'moved back' — a distant cousin that shares the same Latin compound.