The fourth amendment does not protect one from having to show IDs. Unless it is in the pernumbra.
There is no requirement for one to have ANY ID.
Driver's licenses are voluntary, and intended for use only when driving.
SSN cards are voluntary (though hard to avoid), and intended for use only for dealing with the Social Security program; prior inclusion of "not for identification purposes" indicates original intent of non-ID use.
Passports are voluntary, and intended for use only when crossing national borders; majority of people don't have them.
And so on: all IDs are voluntary and for narrow purposes; there is no requirement to possess an ID.
As there is no obligatory National ID, and all IDs are intended for and only required in narrow use, it is preposterous to require - under threat of arrest - that citizens show ID.
As all forms of ID are for specific narrow purposes, demanding ID outside of those purposes amounts to "violating the security of one's papers" per the 4th Amendment. No gov't agent has a right (to wit my arrest if I do not comply) to see my driver's license if I am not driving, my SSN card if I am not addressing Social Security issues, my passport if I am not crossing borders, etc.; the 4th Amendment explicitly states that to see those papers, gov't agents (short of my voluntary compliance, or imminent need) must swear out a search warrant before a judge.
The only time an ID can conceivably be required is if police have reason to believe person A is person B, and B is wanted for criminal activity, police may apprehend A until they confirm A is or is not B, which is made much easier if A produces ID. (This does NOT permit fishing expiditions wherein every person on every bus/train/plane is checked against a list of criminal suspects; they must have articulable reason why they think A may be B; guilty until proven innocent is not acceptable.)