This question is not specific to monkeys. It applies to animals that live with or sufficiently close to human dwellings and would be able to observe our habits daily. I'm not referring to training animals.
Even among animals that have an opposable thumb like ours but different HAR regions, the use of tools seems to be too primitive. Chimps have been known to re-use tools and even cumulative cultural evolution is documented. Whales have been hypothesized to have high intelligence (although not necessarily) due to larger brains containing more spindle neurons. But I couldn't find any info on whether whales use tools (Dolphins are said to use sponges though).
On considering if mirror neurons were helping chimps copy each other and use tools, I found that the very existence of mirror neurons in humans is under question.
So if a chimp can use simple tools, what prevents them from looking at us and imitating our use of something as simple as using a plastic bag or a crude tray to carry a bunch of fruits? Is there a gradation of complexity of tool use that various brains can process? A reason that no matter what happens, a chimp would not be able to imitate our use of tools? Could this be compared to how aged people are unable to learn how to use a smartphone or computer no matter how many times it is explained to them (or even how young people fail to understand pointers and recursion)? Perhaps the ability is related to an animal's capability to form new neural connections of a certain type?
Training an animal like Jack the baboon is different from what I'm asking.