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The official project account recommended against using Aurora Store. Please do your own research.
Apart from that, you are not anonymous or more private while using Aurora Store. Many apps on the Play Store bundle Google Play libraries and those still run on your phone, even if you didn't have Play Services installed. That would theoretically allow Google to link the apps you use to the same user profile anyways. But what threat model does this endanger?
Also apps, including Google Play Services, can't access hardware identifiers.