From Old French 'demener' (to lead) — your demeanor is how you conduct yourself, literally how you lead your life.
Outward behavior or bearing; the way a person presents themselves.
From Middle English 'demeanour' (conduct, management of oneself), from Old French 'se demener' (to conduct oneself, to lead oneself), from 'de-' (intensive prefix) + 'mener' (to lead, to drive), from Latin 'minare' (to drive animals, to threaten, to herd), from 'minae' (threats, menaces, projecting points), from PIE *men- (to project, threaten). The PIE root *men- produced Latin 'mons' (mountain — something that projects), 'minari' (to jut out, threaten), and Greek 'menos' (force, might, spirit). The semantic journey: physical projection (overhanging rock) to threatening gesture (menace) to driving or herding animals by threat to guiding
The word's route from 'driving cattle' through 'leading' to 'conducting oneself' traces the metaphor of self-management.