“[My constituents are] willing to have that discussion about whether marriage needs to be regulated by the state at all.”
Now, the title of the article is "Oklahoma Lawmaker Wants To Ban All Marriages".
IANAL, but banning something is different from unregulating something.
Now, there's only a shell bill, so we don't really know what the tactic is going to be. I expect something like: "OK will no longer issue marriage licenses, but other laws (such as the one that requires officiants to only perform ceremonies for couples with a valid marriage license) remain on the books." That would effectively ban marriage. And I wouldn't expect it to last long, for one reason or another.
But if the law actually did eliminate all state regulation of marriage... and the state is no longer in the business of determining what is and what isn't a valid marriage... then you could marry your refrigerator and your goldfish. OK, it would no longer have any legal meaning within the state, and I imagine(?) the Federal government would not need to recognize a marriage without any state paperwork. But you could do it. Yay liberty?