This subject has come up in various forms over the years…I even started a thread a while back on the idea of having a religious wedding ceremony but just not fill out the civil paperwork…
These things are generally considered because the couple wishes to live together as man and wife, but not be penalized by Social Security.
the general response was that this could not be done in the Catholic Church because the priests are licensed by the state to marry (civilly) and are obligated to report…that is submit paperwork, on anyone they marry.
So they would be breaking the law if they did so.
I can only assume that the protestant ministers who do this are likewise breaking the law.
Now if a couple were to do this, and then seek to become Catholic…I can’t see the Church recognizing the marriage. They would require that the couple get properly married, in the Church which would mean the would be civilly married as well.
Peace
James