From the sounds of things it’s not actually the Android 15 update that’s causing the problems; it’s actually setting up the Private Space feature. At least three users have come across the same issue and posted about their experiences on Reddit. Two of those users had a Pixel 6, while a third had a Pixel 6 Pro.

Android 15 update is causing problems for Google Pixel 6 users, with reports of performance issues and device malfunctions.

It's nasty enough that you should probably hold off on updating. Google finally started rolling out Android 15 to still supported Pixel devices last