That it's been co-opted for marketing isn't really surprising; most of your public records are consumed by private companies to use to make money off you. One big example is how legal proceedings show up on your credit report.
EDIT: Your driving record is indeed a public record. So, easy bet.
Why do companies pay the DMV for this information if they can just FOIA it from them for free?
Technically, yes, you can request records from FOIA and sell them. Though, many public records laws are very specific about whether you are requesting as a for-profit org, and can deny your request on that basis. If that isn't the issue, then your request can be rejected for being voluminous, so generally you have to find a public interest argument for the release of records. That public interest argument very, very likely doesn't exist in the case of the info being sold here. It ends up being a major PITA to get large amounts of data.
So, orgs instead go down the contract route. There's really no need to go through the hassle of public records nuance if your contract includes access to those records.
(This is a super deep subject, so I'm just scratching the surface.)