It could be the copyright for sole purpose of recognizing the author and respecting its ownership of the work, no the copyright for earning profits from sales.
Earth within the Federation may not use money but there are other civilizations and species especially on the outside that still require it to trade - it's possible that humanity just lost the reasons of pursuing acquisition of wealth but still utilizing some kind of currency or just simply replicates whatever its needed to barter with other species. I don't recall that there was any law that required abandoning the currency based economy upon joining the Happy Alpha Quadrant Club - it's possible that member world did follow Earth or other worlds independently in this matter seeing benefits of replication technology. Or it's the natural step of the society evolution of this fictional universe in majority of cases.
It's definitely a really interesting topic that always fuels discussions.
As for the question about AI - I think that granting rights would require first recognizing AI as natural person (or entity) ot legal person and granting them necessary rights; and perhaps this would generate all sorts of other questions, issues as well.
Why do you assume copyright no longer exists in a post-scarcity society?
Patents wouldn't make sense because they are meant as an incentive to share your inventions while protecting your income, but copyright comes with more rights than just money.
Attribution is important, but the episode centers around the question of whether or not the AI can demand that the publisher pull the novel from the market, which is copyright, not just attribution.
