> The opposition argues that this is discriminatory and would favor richer voters
Not quite. They argue that the kind of voter ID requirements that are being passed in several states are discriminatory because they are not free of charge. Some are free of any direct fees, but the indirect costs are often substantial, for the reasons you note:
> If I understand their argument correctly, there is a non-zero cost to going to obtain these forms of identification that disproportionately impact the poor. You have to take off work, commute (potentially without a car or bus), pay a fee, be capable of understanding the instructions, etc.
Worse, some states that have passed such laws have also simultaneously taken steps to increase those indirect costs. E.g., under the guise of budget cutting reducing the number of offices that issue state IDs, and reducing the hours during which the remaining offices processes ID applications. The offices that get closed are disproportionately the ones that are nearest to the most minority and poor voters, who also happen to the the voters who are most likely to not already have the ID, and the reduced hours usually mean no evening or weekend hours meaning many have to take unpaid time off work to go apply for ID.
I forget which state it was, but in a lawsuit over their new ID law plaintiffs found a list the committee that drafted the law made which listed a bunch of different possible forms of ID that people had, and for each also listed for each what percent of voters had it and their party and racial demographics, sorted by how much that ID would favor white people of the majority party in that state, and the final approved IDs in the law were all the ones the ones that favored white majority party voters.
Here are a ton of references, many of which contain a zillion links to even more research on this:
https://www.projectvote.org/wp-content/uploads/2015/06/AMERI...
https://www.politifact.com/factchecks/2012/jul/11/eric-holde...
https://www.aclu.org/documents/oppose-voter-id-legislation-f...
https://www.usccr.gov/files/pubs/2018/Minority_Voting_Access...
https://www.washingtonpost.com/politics/courts_law/getting-a...
https://www.vox.com/xpress/2014/11/4/7157037/us-voter-id-req...
https://www.npr.org/2018/09/07/644648955/for-older-voters-ge...
https://rewirenewsgroup.com/2014/10/16/well-actually-pretty-...
https://www.theregreview.org/2019/01/08/shapiro-moran-burden...
https://www.theatlantic.com/politics/archive/2014/10/heres-h...
https://scholars.org/contribution/high-cost-free-photo-voter...
https://now.tufts.edu/2018/01/23/proving-voter-id-laws-discr...