It seems like there easily could be. What if some of the data they trained it on didn't exist anywhere else except in the training set, and was put there specifically for this purpose? For instance they could have taught it a few poems that don't exist anywhere else. If you can coax the LLM of unknown origin into reciting those poems back to you, you know where it came from.
There's precedent for "whatever you can get away with" in tech companies, but establishing a culture of that at the start of this new big change could end up undesirable for most people.
For example, it could relieve demand for more legal and sustainable ways, until it's too late. (Look at the history of digital entertainment media piracy and DRM and legislation, for example. Or look at the history of software piracy, where some big companies seem to actually want their product to be pirated, partly because it builds a bigger moat against competitors, and they can legally strongarm some of those pirates later.)
Besides: How would anyone ever know which model generated the output you are serving? AFAIK there is no fingerprint in any model’s output. And even if there was, it would probably be destroyed by fine tuning “over it”.