This same question comes up at work (I work for a large company), especially with people born in other countries who are not familiar with American idioms. Many are incredulous that citing the name of an iconic American character, whose cartoon persona always strives to do the right thing and be good, would be used to describe something substandard or below par. I am in my Sixties, but I heard phrases like “Don’t do business with them; everything they do is Mickey Mouse.” (Then there is even the old joke about the Saudi Prince who asked his young son what he most wanted in the world, and the small boy answered, “I really want a Mickey Mouse outfit!”—so his father bought him General Motors…)
I believe the term “Mickey Mouse” originated sometime after the first Mickey Mouse watches, made by Ingersoll, were produced, spawning a large counterfeiting industry of cheap, poorly-made watches featuring the famous character. Unfortunately, import controls were so lax in those days that the knock-offs far outnumbered the originals, so what was once a symbol of quality, that is, having a Mickey Mouse watch, became a term of derision, “Where’d you get that mickey mouse watch, anyway?”—which could then mean ANY cheaply made wristwatch. As these things do, the term took on broader usage to refer to any item or organization that was disorganized or of low quality .
I cannot substantiate any of this or cite any authoritative source, but it is something I recall hearing as a boy back in the 1950s, from my father.