I assume the couple get a piece of paper that shows that they met the requirements.
I assume that if the church is allowed to issue a piece of paper, that the paper would have to be filed with the government, rather than in each individual church or Muslim mosque, or Hindu Temple, and atheist don’t have churches anyway.
Paper will exist, and the purpose of licensing will still exist, there have to be records.
Each couple,could,keep the paper themselves and produce it if necessary.