No, the FIRST monkey to write Shakespeare used a feather and ink.

It only took a couple hundred years after all those millions for them to be written on the typewriter.

It's a thought experiment, not an observation. The idea is that if you have infinity and it's truly random than eventually all possibilities emerge somewhere within that.

The idea of infinite monkeys typing randomly on infinite typewriters is that eventually one of them would accidentally type out all the works of Shakespeare. Many more would type out parts of the works of Shakespeare. And many many many more would type random garbage.

If we then take that forwadd imagine for a moment the multiverse is also infinite and random, then every possible universe would exist somewhere in that multiverse.

It can be taken in other directions too. It's a way of cocneptualising the implications of infinity and true randomness.

Meanwhile actual Shakespeare had intelligence and wrote and created his works. Him being a monkey writing Shakespeare is just a sly humerous observation, but it has nothing to do with the actual meaning of the thought experiment and the idea it is trying to convey.