My understanding is that weddings “by declaration” – that is, couples exchanging vows in front of witnesses, without the need for a priest – were also legal in much of Europe in the Middle Ages.
Marriage customs are an interesting area to play with in your fiction. In my Gryphon Clerks novels, the requirements are two witnesses who have taken adulthood rites, a “person of standing” to conduct the ceremony, and filling in a simple form.
In one of the core novels of the evolution of Italian language, I Promessi Sposi ( https://en.wikipedia.org/wiki/The_Betrothed_%28Manzoni_novel ), the titular couple tries something like that, but in front of the priest.
The priest, under blackmail of the local rich & powerful guy, stalls the couple’s wedding. The couple tries a wedding by declaration, trying to surprise the priest and having him listen to their declaration, believing that this will force him to acknowledge their marriage.
We don’t know how this turns out legally because they fail.
I think the law gradually required more rigour around the ceremony, and the church also carved out a bigger role for itself.