In Ohio, it's much easier to vote with a drivers license than with a state ID card. People that are disabled or can't afford a car have state ID cards.
Also, poll taxes are unconstitutional in the US. $10 is more than $0.
It takes very little to imagine how this causes inequality. Maybe you can’t get the state ID either! Voter ID and State ID aren’t necessarily the same thing. Maybe you need both! Maybe you can get your State ID at a local office but the Voter ID only from the county courthouse two towns away. Maybe you don’t have a car and a day off. Maybe they are only available on certain days and times. Maybe those times change at the last minute.
There's a big can of worms here. The thumbnail sketch is "Americans have some (as viewed from outside the US) odd and severe hangups about being tracked by the government that is, ostensibly, theirs."
Reasons range from the practical / legal ones listed by the ACLU (https://www.aclu.org/other/5-problems-national-id-cards) to a small-but-vocal subset of voters who actually believe (because so much of the US is descended from Christian zealots fleeing persecution in their home countries for heterodoxy) that a card issued by your government that is required to participate in society is a literal "mark of the beast" as per the biblical Book of Revelations and therefore something to be resisted as part of a struggle against anti-Christendom.
Also, poll taxes are unconstitutional in the US. $10 is more than $0.