Each copy of a luxury provides amenities to four cities. If you have 5-8 cities, then you'll need two copies to give one amenity to each city.
Pretty sure it's one type of luxury for 4 cities, no matter how many copies you have. But that's typically the type of things that should be clearly mentioned in the civilopedia.
It is just 4 amenities from a unique luxury. Extra copies of that luxury provide no additional amenities, regardless of how many cities you have.
As for the question posed by the OP (why does the AI seem to want to buy extra luxuries?), that is probably an AI trading glitch, and not any indication that somehow extra luxuries actually have value to the AI.