First of all, that's not branding, it's bias for a known, existing solution.
Second of all, people have switched browsers en masse before, it just requires a good enough reason to do so. People switched from IE6 to Firefox in a relatively large number, and then they all switched to Chrome. Even today, people on Windows tend to install Chrome, while people on Macs tend to stick to Safari, so some browser choice is happening.
Third of all, and perhaps most importantly really, Chrome is free, so the only reason to switch away would be finding some major features you want in another browser. ChatGPT is not free, and it's losing money fast at its current price point. But if they make it more expensive, as they need to unless they can significantly reduce their costs, people will look for alternatives. Other than fashion brands, I don't think you'll find almost any example of people in general using a more expensive service when a cheaper equivalent one exists and is being actively marketed.