The LLM is not employed by you or your employer, because you can't enter contracts with non human or non human organizations.
When you license a non-LLM code generation service (like a page that creates a website for you), that company owns the copyright of the generated website because their deterministic system generated code by defined rules and mechanisms that were defined by the code generation system. Assuming no LLM as part of that, there is no code that is generated by the system outside of the rules that they defined (it's not filling in the blanks that you or the code generation system didn't explicitly define).
Since they own the copyright of the website, they can then assign the copyright and authorship to you because of your license agreement to them.
Since the LLM is filling in the blanks on its own in undefined ways, it is the author and not Anthropic/OpenAI/ETC. That means that even though you have a license agreement with Anthropic/OpenAI/etc.. to transfer copyright, they didn't have copyright/authorship, the LLM did. And since the LLM can't legally own copyright/authorship (since it isn't a human) then it can't grant it to you and you can't then grant it to your employer.
In reality I fear it's going to cause less copyright protections even for small developers.
I think what this means is that the employee may not be the copyright owner for multiple reasons, which are possibly applicable simultaneously. It does not imply that the employer owns copyright over the work that is in public domain, which would be a contradiction.
Under at least EU AI Act, any work done by AI is not granted copyright. But it does not mean copyright does not apply, it means the amount of work credited to AI is set at 0% (simplification). A human working off another's work unless it's perfect copy will have "credit" for changes that are judged creative/transformative, meaning a human plagiarizing something still can claim to have some degree of authorship. An AI won't.
In a sense, the copyright status of final work is a sort of "sum with dilution" were each work involved adds to claims, but AI's output is set at 0 - the prompt or further rework by human is not.
As for employer, details vary but generally "work for hire" rules and contracts do reassignment of material rights (in EU and some other places you can not reassign moral rights which are a different thing).
