I think we will see a hidden motivation behind this as ultimately so the output can have an author, the author can be attributed and finally the output will be copyrighted and so the LLM has more value.
Everyone who invests in AI companies wants to see the value of their investment increase.
I'd give it about 3-5 years until an AI company claims copyright over code their LLM produces. This is a crucial step in that path.
I think you're right that it's actually to do with attribution (i.e. They're not just watermarking your output as claude generated, but watermarking it as claude generated _by claude user id 73684_). I think it's more to do with the growing militarization of the internet. Just another brick in the wall of enter your phone number to create an account, send your id to prove your age, smile for the flock cameras, etc. Allowing the plebs to have privacy and anonymity is not allowed anymore.
As someone who doesn't utilize LLMs for writing production code I'm looking forward to starting a consultancy and reaping the rewards.