This is why marriage is actually the fairest system - it doesn't matter if you live together (lots of married couples can't full time for work reasons but remain married), it doesn't matter if you have children, it doesn't matter if you're sexually active or chaste, it doesn't matter if you have joint property, it doesn't matter if you've been together for ten years or ten weeks - once you've said the magic words you're legally linked as much as anyone else.
Big practical problems here (aside from the fact that it's utterly stupid and superfluous when marriage exists)
How are you going to differentiate people who are an "unmarried couple" and who are flatmates? Maybe some sort of... registration?
What happens if a couple splits up and one person just denies they were actually a couple?
What happens if your flatmate of 3 years gets hit by a bus and cops it- do you just get all their stuff by default?