Yes, Google does record and store unique hardware identifiers of devices signed into Google accounts via Google Play Services. I think it includes hardware MAC address, but I am not sure. The only sensible way to sign into Google account is by using a browser like Vanadium, Brave, Bromite, Firefox or Tor Browser.
It makes me feel really bad for people who purchase VPN subscriptions and sign into Google with Google Play Services over those VPN subscriptions. Sure, VPN hides some network information, but hardware ID's of your phone get sent to Google...