In the page I linked above, Anthropic explains that they use the techniques from SynthID-Text, where a secret watermarking key is involved. Only Anthropic knows this key, so only Anthropic can produce and detect the watermark. (But they also promised to provide an API through which other parties can check the watermark.)

SynthID-Text paper: https://www.nature.com/articles/s41586-024-08025-4