You are correct. The author of the article needs to go to remedial journalism school. The actual report from Public Interest Legal Foundation states that the problem is NY's failure to maintain the ID data provided by the voters in the NY state database. New York is required by federal law to do so, but apparently did not.
It isn't voters without IDs, it is a voter registration database missing the ID numbers for millions of voters. That's a failure by the New York government.