It's because Americans have a libertarian streak and instinctively mistrust the government, so we've historically resisted any sort of federal ID program. Because the federal government has to keep track of you somehow, and every American (more or less) already had a unique identifying number, the government started using those unique identifying numbers to identify us for tax purposes. This started being a convenient way for private corporations to ask the government "Wait, who's supposed to be identified by this number?" for employment and loans, and then people decided that it would be better to use this instead of any sort of, you know, online ID, to identify people for credit card applications.
I'm British, we've also resisted it, which is why the closes thing to a national identity document I listed was a driving licence or passport. It's not uncommon to have a provisional driving licence (learners' permit to you I think) purely for the purpose of being IDd for alcohol or whatever.
For tax I would use my national insurance number and any form of photo ID to register for an online account, or you can do a paper form but I don't know how they verify identity if you're claiming a refund in that case.
A credit card application would similarly want a copy of photo ID and probably a proof of address (like a bank statement or utility bill).
There's nothing wrong with having the number, I have a national insurance number, I have Self-Assessment Tax Return numbers (I think it's unique each year), it's the secret bit I don't get.
Remember, our drivers licenses are issued by the states, not the feds. That means there's 51 competing standards, and not allot them conform to the requirements of the feds (hence the slow, partial roll-out of RealID)
The lack of a federal ID is almost irrelevant at this point. Most states have moved to "Real ID" which requires almost as much if not the same to get as a passport (federal ID). Requirement to have that has been coming (and admittedly pushed back several times) for a long time.
It was also heavily rejected as a "number of the beast" by far right Christians -- who are consistently able to get guys like GW Bush and Trump elected.
No one on the left really cares that much -- they have their own sacred cows and causes of the year -- so nothing really changes.
