There is the semantic issue, and then there is taxonomic issue. The later of which involves cladistics.
Even within Europe many languages don't distinquish "monkey" from "ape". And ape originally mean just "a tailless monkey". The Barbary "apes" of Gibralter and N. Africa are really tailess macaque monkeys. But they were so named centuries before Europeans even encountered the animals that English speakers now call "apes" (chimps, gorillas, bonobos, orangutans, gibbons). And (as with Wolfram's Swedish) most nonscientists English speakers kinda use ape and monkey interchangeably anyhow.
But lets say you go by that distinction: apes are what we now call "apes", and "monkeys" (including tailless ones) are monkeys. Then- strictly speaking- humans would be apes. And not "monkeys".
Except wouldn't humans...also....be monkeys? Humans evolved from the apes, and didn't the apes in turn evolve from old world monkeys?
That is one assumption. If so then yes we would be both Apes and (with the rest of the apes) we would also be old world monkeys.
Except we are not sure if human ancestry actually went through a monkey stage.
Monkeys themselves evolved from lemurs. But there are a number of primate species that are not lemurs, like the bush baby, the tarsier, the loris. Some experts say that apes evolved from tarsiers, separately from monkeys. If that were the case then we would not be monkeys even though we are apes.
But most experts place apes in the same clad as old world monkeys, and say that old and new world monkeys have a recent a very monkey like ancestor. And apes (and us) being within the old world monkey clad we then be..old world monkeys.
Fun fact: to distinquish the modern categories of "monkey" and "ape" tail status is not as important as which kind of joint the animal has in its shoulders. We humans can fully extend our arms sideways (like Christ on the cross). Most mammals cant do that because the joint at the top of their fore limbs only rotates in one direction. Dogs and cats cant extend their arms sideways, and can only move their fore legs in the front-to back plane.
Same with horses. And the same with most mammals, and even with most primates.. Including "monkeys". Capuchin monkeys of South America (New World) and macaques of the old world can only move their otherwise quite human like arms and hands the same way a dog and cat can move them (front to back), and they cannot pose like Christ anymore than a horse can. But all apes (chimps, gorillas, orangs, gibbons) have universal joints in their shoulders like we do, and can extend their arms out sideways in a Christ pose as well as we can.