There's a computerphile video (https://www.youtube.com/watch?v=XZJc1p6RE78) with Dr. Mark Pound explaining a paper by John Kirchenbauer, Jonas Geiping et al. (https://arxiv.org/abs/2301.10226) that described a method for watermarking LLM output like this. It's not directly stated anywhere in the Claude support article that this is what they're using, but the properties of the watermark described seem to point to this method.
The bias has to be small enough that if you ask an LLM to repeat some passage of text like the national anthem, either from the training data or from the prompt it doesn't change random words.
Based on my understanding, it can only be applied to code in very limited ways: docstrings, variable names, string literals. The code itself can't really have tokens changed to another equally correct token (the foundation of the watermark) because then the code breaks! And the few places that you can do so are likely erased by formatters anyway.