It is generally the state’s responsibility to maintain and issue identification cards and liscenses. The only overarching American identification is your passport and Social Security number, both of which are technically voluntary. The Real ID act you might have heard about in the news recently is supposed to more closely unify the various liscenses into a similar format for air travel and voting compliance, but many still do not have this new form of ID.
America created the Social Security number and kind of gave up.
The most simple answer: We wanted as many Germans to eventually use against the Soviets. Anyone who was not a big name was kept around for the eventual return of the West German Military. Same with scientists and operation paperclip. The soviets did the same to eventually be used against the rest of Europe.