I vote no on both options. Monique is very unlikely to morph into monkey. There are a lot of monkeys in English (monkeyshines and so forth) slang. Not sure what made English speakers think of monkeys in making slang. Seems to be several things that added monkey to make a phrase. Monkeyshines reflected the seeming foolish activities of the creatures. Grease monkeys worked up in the rafters to keep the old line-shaft shops lubricated. Monkey-tail cannons were small ship's cannons (couple of feet long or so) that had a tiller attached to the rear that reminded people of the tails of the critters.
Maybe we should monkey around to find the earliest reference to monkey wrenches.