You can get the recipe for a wedding ring from the travelling cart while in co-op, costing 5 iridium bars and one prismatic shard.
it only lets you marry another player, not any villager.
This is, what the Wiki says about marrying in co-op, you can marry your co-op Partner and have children , but there's nothing said about dating, unfortunately, at least no cutscenes or heart-events... That's up to you, maybe you can create your own with your partner could be interesting
"Marriage
Players can marry an NPC, which works the same way as single-player. Each NPC can only be married by one player at a time, and children are added to the married player's home.
Players can marry another player by giving them a Wedding Ring. Married players get a Stardrop in a purple giftbox next to their home beds after the wedding (equivalent to the Stardrop received from an NPC spouse). A home is eligible for children if it's owned by one of the married players, has the nursery upgrade, doesn't already have two children, and both players are sleeping in its bed; the game will randomly let one player propose having children to the other, and if accepted, the child will be added to the home that both players are sleeping in (if eligible), else the other player's home (if eligible), else the current home. "