I would say perhaps a more apropos example would be needing a photo ID for flying when you go through TSA.
...and why I can't just show up at the airport with my voter registration card and say "See, I'm Rob, it's written right there on the card"
The point of a photo ID card, and it being a card that was issued by the state/federal government (and not just something a person filled out) is so that a person in charge of screening (for whatever purpose) can look at it, and have some assurances that the person handing them the card is actually the person referenced on the card.
Otherwise, there is literally nothing that would stop me from voting on behalf of a family member who was roughly the same age, who I knew wasn't planning on voting.
And it's not even a controversial practice, a large plurality of voters, across nearly every demographic category, is highly in favor of it.
How some states can have ID-free voting in conjunction with universal mail-in ballots, and still try to assert "our elections are completely secure and there's no evidence there's anything to be concerned about" is counterintuitive.