The last meaning listed above is the meaning being used. I would define it this way:

prop (B): an object within a staged theater, movie, photographic, or literary scene that supports the visual representation of the created context and that actors within the scene may use to advance the realistic representation of dramatic activity (as in a stage play where one actor strikes another with a candlestick) or to enhance a representational or suggestive pose (as in a still photograph where the subject holds a musical instrument or, perhaps, a flower).

The "props department" is a common phrase used to refer to the organizational entity within a theatrical company, television production studio, movie studio, or advertising agency that is in charge of all the things that might be used to stage a scene from costuming to furniture.

The etymology is derived from the primary meaning of the word (for some reason apparently completely overlooked by Wiktionary):

prop (A): something that supports, sustains, or otherwise 'props up.'

"Support" is the primary synonym. The props in a staged scene support the illusion of reality.

False etymologies claim this meaning to be a shortening of "property," but this is incorrect. Property comes into the language from Latin. "Prop" is an Old English, Germanic word related to the concept of a "stopper" as an object that prevents one or more other objects or substances from advancing, collapsing, or otherwise moving in undesirable ways.