My (admittedly superficial) understanding of the problem is that it's about making it compulsory to have an ID to vote, more than about the ID itself (which de facto exists for nearly all americans, in the form of a social security number).
This means an extra administrative step on the road to voting, and an opportunity to add any number of potholes along the way for minorities an administration doesn't want to see voting.
Keep in mind that the southern US states are a place where revisionism about the civil war, racial segregation, and the KKK were all a thing. As an example of pothole that already exists today, one issue is that voting in the US can occur during a week day, and it's not beneath some employers to disallow part or all of their staff to take an hour or two off to vote.
For an example closer to home, picture laws designed to infringe this or that minority or caste's right to vote and run for elections.