Device stores MAC address of each Bluetooth contact it makes. When new infection confirmed, all devices are pinged and those who have stored the address of the new patient’s device get a notification to consider self-isolating because they’ve been in touch with a confirmed coronavirus bearer.
It’s not more than that. A basic notification. There’s no identification at work at all. But we won’t know until both roll out their implementations.
As for your other thoughts, of course that happens and has happened for quite a while already. And on massive scale. At the same time there’s governments (and populations) who object and create stringent privacy laws allowing only approved use of such information. Not Trump tho.