It's absolutely not true. People have to supply IDs for tons of activities. They have IDs. We know who they are. They are registered to vote -- how did that happen w/o ID? Of course they have IDs.
Where do you get this idea that you need to have an ID card in order to register to vote? It's certainly not a federal requirement.
In NY, you can register with ID, last 4 digits of your social, or leave it blank. If you leave it blank, you will need to provide some sort of identification when voting, but a utility bill in your name and address will suffice.
The statistics just don't back this up. Plenty of, predominantly poor, people don't have driver's licenses. And that's typically the only ID people have. Also, poorer people may work under the table or deal in cash.