I'm looking for some details on the development of feminine French mode from masculine Latin modus. Why was there a change in gender here? Both masculine mode and feminine mode mean more or less "manner", so what's the point of assigning femininity to mode? Is it possible that the feminine gender was by analogy with manière or façon?
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 This masculine mode superseded the old masculine meuf, not to confuse with the new verlan meuf ;-)
1While 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.