I think these things are mainly based on cookie/fingerprinting these days - the check-box is just there for show. People like cloudflare and google get to see a big chunk of browsing activity for the entire planet, so they can see if the activity coming from an IP/Browser looks "bot like" or not.

I have never used ChatGPT so no idea how its agent works, but if it is driving your browser directly then it will look like you. If it is coming from some random IP address from a VM in Azure or AWS even then the activity probably does not look "bot like" since it is doing agentic things and so acting quite like a human I expect.