Whether or not ChatGPT can detect ChatGPT-style text depends on whether or not ChatGPT can be taught to recognise that style.
Humans can often spot some common patterns in ChatGPT responses, and ChatGPT can be taught to do detect patterns, so it should be possible to teach the patterns to ChatGPT.
In conclusion, ChatGPT should be able to detect ChatGPT-style text.