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.
My point was that they could make a compelling case though, not that they would win.
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.
Who owns the copyright for something not written by anybody, you ask? Is it the man who pays to have it written, or the owner of the machine that does the writing? But it is neither. Nobody owns the copyright because nobody has written it.
Anthropic's user agreement does not have a similar agreement.