mode (n.1)

"manner," late 14c., "kind of musical scale," from Latin modus "measure, extent, quantity; proper measure, rhythm, song; a way, manner, fashion, style" (in Late Latin also "mood" in grammar and logic), from PIE root *med- "take appropriate measures." Meaning "manner in which a thing is done" first recorded 1660s.

mode (n.2)

"current fashion," 1640s, from French mode "manner, fashion, style" (15c.), from Latin modus "a way, manner, fashion, style," from PIE root *med- "take appropriate measures."