In a system like the US, I think it's better for the states to handle identification cards, not the federal government. There's a lot of details that go into identifying someone, and your local state is best equipped to handle that.
To give a few examples:
What's the procedure for changing your name? A lot of people change their name after marriage or divorce. Those are usually handled by the states, so adding a federal government in there raises the net complexity.
Should people in Alabama be able to force the residents of California to put their original sex on a federal id card? I don't care, but other people do. It could be one of those pointless political distractions that's better left unanswered.
If the federal government runs it, now they have to own or rent a bunch of land in every major city in every state so they can service id card requests. So it gives them a larger presence. I guess they could reuse social security offices, but they couldn't reuse state DMV offices which are the more natural choice.
Don't we already have that with the post offices and the passport system?
Couldn't they expand passports into a Federal ID system?
I'm not qualified to comment on constitutional issues so I won't.
I will say that, even if it's mostly an emotional issue (people already need at least some sort of state government-issued ID to do lots of things), there's a huge resistance in the US to anything that smells like "papers, please" or universal ID that people are required to carry and present upon request.
Which is how a lot of people view the idea of a Federal ID system. That may seem silly to a lot of people outside the US but it's how things are.
And yet, you are, under certain circumstances, already required to ID yourself to a police officer upon request.[0] The most common form of ID? Your driver's license.
[0]: https://en.wikipedia.org/wiki/Stop_and_identify_statutes
The Tenth Amendment's Reserved Powers clause also suggests this is a states issue, unless you can contrive it to be part of the census.
To be clear, the federal government is already involved with name changes-- in fact, though my name change was issued by a state court, I had to get it changed with Social Security before I could get a new driver's license.
Similarly, passports do have a sex listed, so that's already something with federal jurisdiction. The passport system is also handled through post offices, solving the problem of needing more offices.
Fifty cryptographic APIs, on the other hand, would be a nightmare. We'd at least need the federal government to force states to implement one conforming to some open standard so that the integrations aren't intractable.