There is no mandated form of ID in any state. How can you require people to have something to vote that is not required for any other purpose, at least if the government is not required to provide it?
In nations that do in fact have a national ID, everyone gets one, with little or no effort (certainly no need to travel), and if you wanted to mark the card "non-voting" that wouldn't be a big step.
Your right to vote is established by the US Constitution. A state can't burden that right. If a state wants to add an ID requirement, that's fine, but getting the ID must present no burden to any voter.
In reality, the conundrum here is entirely of conservatives' own making. On the one hand, they (non-exclusively) are adamantly against automatically issued national or even state IDs. On the other, they want to require ID for voting (ascribe whatever reason to this you want). You can't fulfill both these desires without violating the constitutional right to vote.
If you are required to have documents and paperwork for a voter ID that you are not otherwise required to have, that is, IMO, an undue burden.
I have no problem with the state saying "you must have this document in order to vote" (it even makes a limited amount of sense, despite being a solution in search of a problem). I do have a problem with the state doing anything other than ensuring that every resident who should that document gets it with minimal effort (preferably zero).