ChatGPT isn't there yet, but I can see it replacing Google search in the future. Just like Linux never won the desktop, but Android replaced Windows by numbers, it's quite possible that the old way we search the open web will be replaced by a different model. An AI that answers all your questions. The timing is good, because Google search is getting worse every year, mostly due to SEO spam. Having tools like ChatGPT feels like SEO spam will get even easier to generate and harder to detect.
ChatGPT plans to charge for usage, and that could be their biggest mistake. If I were Google, I would jump on this, make my own version and give it away for free until all competition runs out of money. Gradually add ads on top of it and keep the leading position. They can even use their dominant position with Android and Chrome to make their AI the default "AI engine".
If you feel like they stand no chance, remember that Internet Explorer won and used to be the default for years and years.