I think it's prompted with a bunch of context information (like, "you are a helpful, harmless, honest AI assistant created by OpenAI, your knowledge cutoff date is ..., please answer the user's questions").

If you really think it is just saying whatever it read on the web, how do you explain that not all LLM chatbots claim to be ChatGPT?

Engineering is happening, it's not just a raw model of text from the web connected directly to the user.