The etymology is quoting only two of of the major meanings of “monter” and missing what is here the key one: “to assemble, construct” (sense II.B. of the Trésor de la Langue Française), though I’m not entirely clear how the derivation between the original meaning (“go up”) and this one occurred. Possibly through the use of “monter” for structures being built?
Montage comes from monter. The meaning of monter has widened from “go up” (from late latin montare) to: “assemble, joining different parts of a mechanism so that it could be used” (used at first in clockmaking and printing in the 16th century).
From there derives the figurative meaning of “organise, combine together, assemble” dating back to the 18th century. It was first used for a theatre play, hence montage which has been taken over by the cinema industry to describe the selection and arrangement of separate cinematographic shots to form a single picture.
From this use, on the stage, it further evolved to being used in the arts (e.g. visual arts, music) to describe the process of producing a single work by combining several different elements.
Dates and part of the information is condensed from Dictionnaire Historique de la Langue Française (Alain Rey & al).