Nobody "owns" it for the most part. Most LLM output can not be copyrighted, because there is not enough human essence in it. Meaning that it is public domain the moment it is produced.

There might be a few exceptions, if you really guided your model in a certain particular way, but that is rare.

What /could/ be is that you violate their ToS. If that ToS is valid and legally and practically enforceable is a different question and will depend a lot on your jurisdiction.