You don't actually believe that a non-id database would actually get set up, do you? And the ATF isn't supposed to be making a list of gun owners either, but it is.
As for the database, if no identifying information may be collected at the point of entry, how would it get in there? There would, of course, need to be a secure system for making entries into the database, but I think it's doable. There are only three ways of becoming a citizen: being born here, being born to an American citizen abroad, or being naturalized. Some sort of short-term card could be issued at hospitals and naturalization centers, authorizing the holder to be entered into the database within a short period (a few days).
As long as the entry process occurs in groups, personal information can't be attached to the biometric data -- e.g. you can show up at any entry center and surrender the card (which would contain minimal identifying information, like approximate age, gender, race, but no specific identifying information like a name) in order to get in and then go into a room for biometric entry 20, 30, 50 people at a time in random order, with no presentation of any personal information before putting in your biometric data. Some sort of non-identifying provision would need to made for infants, who would obviously have to be accompanied by an adult who would not necessarily be eligible for entry into the database -- perhaps upon surrendering the card, each person or guardian of an infant would be given a tag of some sort indicating whether the holder or an infant accompanied by the holder is eligible for entry.
The initial set-up of such a system would be a bureacratic nightmare, though no more so than for a national ID system which does include personal identification; but after that I think it would be very manageable.