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My guess is that many ancient instruments would be custom ordered from a luthier or other craftsman who specializes in a specific group of instruments. It would help to know which instrument you are looking for or which region it comes from.
First of all, what’s “ancient”? As in “Ancient Greece”? Assuming you meant “historical” instead:
People do. Not companies. Why do you think a Stradivari violin is called that? Because Stradivari built it (with some help no doubt). So if you want an instrument like that, it stands to reason that you use the same process.
I just learned from another comment that these people are called luthiers in English. Some still exist.
Fender.
The hurdy gurdy is certainly an ancient instrument. I have become enamored of this kit maker.
Zildjian is such a an old company still making the same thing. I don’t think cymbals changed much. People sell lutes on Etsy, those are an old style stringed instrument and flutes are so old too.
Lyon & Healy