Because shared understanding of what makes for a numerically identical consciousness is the bedrock of much of human society.
Am I the same person I was twenty years ago? In many ways no. Superficially I’m balder and creakier. Less superficially I have very different ideas about everything from relationships to politics. Put me in a situation I was in 10 years ago and I would likely choose very differently.
Yadda yadda yadda. Am I the same person who signed for a mortgage twenty years ago? Oh hell yes. All the qualitative changes in the world won’t let me wriggle out of that one. The guy who signed and the guy who’s paying now are one and the same. Same with non-legal agreements. Same with relationships. I am the me of twenty or ten or five or one year ago and if we were all to start acting like that wasn’t true it would be anarchy.
So, let’s put some money on the table: You, @Sam_Stone, find a magic ring that purports to duplicate the wearer. Being cautious, you want to test it in a human guinea pig. Being reckless, I agree to be your testee for $1000. We sign a contract to that effect. I put on the ring. There is a blinding flash and when you can see clearly again lo! there are two @Stanislauses before you.
“Well”, says the one on the left, “That seemed to work fine. I’ll take the money in cash if that’s ok with you.”
“Woah there handsome,” says the one on the right. “I’m the one who signed the contract. I’ll take the cash.”
“Don’t talk nonsense you good-looking fool,” says Lefty. “I have a very clear memory of signing the contract, I did it then I put on the ring. I’m the one who gets paid.”
So, who do you pay? Both of them will swear on oath that they have a continuous consciousness from signature to duplication. You don’t know exactly how the ring works so you don’t know which if either is physically continuous with the contract signer. Do you
a) pay neither of them. The contract signing @Stanislaus ceased to exist when he put on the ring and neither of these guys in front of you now made any deal with you.
b) pay one of them $1000 dollars. One of them is the contract signer in exactly the same way that you are the guy who signed your last contract. You did it then, you kept existing, you are that same guy now. Same here.
c) Pay both of them $500. They are both jointly the person who signed the contract and that person, though now bifurcated, has a legitimate claim to $1000.
d) Pay both of them $1000. They are both separately the person who signed the contract and thus they both have a legitimate claim to $1000.
Of course, option b) comes with the extra wrinkle that you have to decide which one is numerically identical with the contract signer. How would you do that?