I don't know that means Bluetooth will be running when the device is off. "Specialized hardware" could mean a full Bluetooth modem on backup power, but more likely it's means there's a low power beacon. Would be interesting if anyone does a teardown of the Pixel 8.

For non-Pixel 8 devices, definitely not. I assume "Offline" refers to the case where your device doesn't have WiFi/LTE, but can still use Bluetooth to communicate with devices that do.