Downloading apps from the Aurora Store allows you to download apps from the Google Play Store without needing the Play Store, a Google account or Google Play Services. So using Aurora Store allows you to download apps anonymously.
As for the second question, I'm not sure but I don't think so. Google, however, will know what apps you download. Of course you can make a Google account at least somewhat anonymously (fake name, use a VPN, burner phone number, alias email, etc) if you wish to use the Google Play Store which does have better security than the Aurora Store.