Earlier quoted context omitted.
Anthropic could at least make a compelling case for the copyright. It becomes legally challenging with regards to ownership if I ever use work equipment for a personal project. If it later takes off they could very well try to claim ownership in its entirety simply because I ran a test once (yes, there's a while silicon valley season for it). I don't know if they'd win, but Anthropic absolutely would be able to claim…
Using work equipment for a personal project only matters because you signed a contract giving all of your IP to your employer for anything you did with (or sometimes without) your employer's equipment. Anthropic's user agreement does not have a similar agreement.
I don't know of ant precedent where the code was literally generated on someone else's system. Its an open question whether that implies any legal right to the work and I could pretty easily see a court accepting the case.