It doesn't appear to be an attempt at plural (obviously that'd be wrong in any case), and possessive doesn't make any sense in either this, or the common use, context.
Based on https://www.insider.com/what-drivers-license-looks-like-in-e...:
17 states say "Driver's License". Arkansas, Georgia, Idaho, Illinois, Kansas, Kentucky, Louisiana, Maine, Maryland, Massachusetts, Minnesota, New Mexico, Pennsylvania, South Carolina, Vermont, Virginia, West Virginia.
Indiana and South Dakota actually call it an "Operator License", rather than "Driver" or "Driver's".
Did you count the other 33 as all having 'Driver license'?
Not that that's terribly conclusive -- it looks like each licence design is performed without the benefits of an experienced sub-editor.
I always assumed that the possessive form was to indicate that it was a license possessed by a driver. In fact “Driver License” makes less sense in my head, as I can’t make heads or tails of what role each word is supposed to make in that clause.