Yeah they do register it. All merchants do. But do you care? I mean if you want to be off the grid because you get targeted by nation states it may ne problematic. If you want google to not show you ads, or obtain data i don’t think it matters.
Once again thread model matters.
They don’t track, it, they would keep a copy of the serial though and invoice as would any shop that sells mobile phones. Your telephone carrier will also know your IMEI, IMSI, as well, as soon as you activate your connection. Most countries require identification for service.
This really isn’t an issue though as it’s not exposed in most applications, eg browsers and requires the very highly protected READ_PRIVILEGED_PHONE_STATE permission. Google Apps, and Micro G have this permission, but they aren’t web browsers, and would use it for eSIM registration.
TLDR i wouldn’t worry yourself about it. Yes, Google knows you have a Google phone, but not much else. They would also know your phone number from the invoice, but then they would also know that if you used Google Messages too, which uses RCS to other Android users.
One could argue that’s a privacy improvement as their RCS implementation in Google Messages uses the Signal Protocol, which is an improvement over unencrypted SMS transmission.
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