Can American governments require citizens to carry documentation?

True. For some reason there’s common thinking that they are indivisible.

Already my employers for most of my adult life have been required to verify they saw my Social Security Card and some form of ID (and if a noncitizen, proof of employable status such as a visa or the so-called “Green Card”). That is considered a reasonable burden as necessary for purposes of processing payroll and income tax and whatever labor benefits may apply to the employee.
Of course, I only have to present that at the time of inprocessing, and they just fill a checkbox next to a line that reads “presented [document name] #ASDFGHJK issued by agency XYZ”

Extrapolating a bit further…
Let us imagine a single standard universal ID document. I do not believe that its mere existence would necessarily require the passage of (a) an exclusive mandate for the person to use it and nothing else in any official dealings; (b) a mandate to have it on your person at all times and failure to do so be a per-se offense (as opposed to a requirement to produce it for cause, on reasonable demand).

Yes, I know… “why would you create a single universal ID if you are not going to compel people to always have it?” Fine, so let’s say it’s so that poor minorities can have a voter ID and no county clerk may hassle them for using it for that. So that uninformed clerks will not think your New Mexico/Hawaii/Puerto Rico driving license is “foreign”. Surely the only purpose cannot be nefarious.