Le mode and la mode indeed share the same root but they are specialized so you cannot use one for another.

Despite modus being masculine in Latin, the gender of mode was initially essentially feminine for an unknown reason¹. In parallel, there was also the masculine meuf/mœuf also coming from the Latin modus which was used in grammar and music (mood).

Later, the masculine mode was reintroduced to agree with the Latin and distinguish “technical” use (masculine) from “non technical” (feminine). This masculine mode superseded the old masculine meuf, not to confuse with the new verlan meuf 😉

¹_{While very rare, word gender change during the evolution from Latin to French did happen. For example the feminine la fin comes from the masculine Latin finis. Italian il fine and Spanish el fin kept the Latin gender. La fleur fem. comes from Latin flos/floris. Italian has il fiore masc. but Spanish has la flor fem.}