"This is a copy; the original painting is in the Louvre museum". Why can't we use the word "genuine" (not a copy or a fake), instead of "original?"
The word original works better when there is only one work in question:
This is a copy; the original Mona Lisa hangs in the Louvre.
The word genuine could work if we are talking about something that is not a forgery or a reproduction:
This is a genuine Manet painting.
means that Manet painted it; it's not some facsimile of a Manet work.
In some contexts, either word could be used:
His whole life, he saved money, hoping one day to buy a genuine Picasso.
His whole life, he saved money, hoping one day to buy an original Picasso.
Both of those refer to a work created by Picasso's hand, although the first carries a slight connotation of "as opposed to a replica."
'Genuine' in this case (the world of fine art) would imply that the copy is a fake, a forgery, intended to deceive or defraud a buyer.
Instead, it was probably a copy by hand or photographic reproduction intended to illustrate the work for audiences who could not travel to the Louvre. So we refer to the "original" work of art.
In this case, "original" and "copy" are part of the jargon used. The original is that which the copy is made from.
Compare with using a copying machine. If you copy a document, the document being duplicated is referred to as "the original" and the duplicate is "the copy". Even if "the original" may itself be a copy of the real, genuine document!