Keeping legality aside, what is the problem in sharing a copy of your resignation letter ? This can be one of the only ways the hiring manager can know that you have surely started your process of leaving the other company.
Is this normal?
Based on other questions on this site I can almost see why its normal in Asian countries to do this. Although in my opinion it doesn't really make any sense as you could always pull back your resignation and it isn't really hard to fudge an already sent resignation email.
So, although strange, I think this really could be the procedure your hiring manager follows and I don't think there is harm in sharing your resignation letter.
I thought the resignation letter is confidential. Is it not?
I have never heard of a resignation letter being confidential. The process of resigning itself may be confidential but only for a couple of days. It isn't uncommon for a manager to announce to his team that X is leaving and the transition plans are in place etc.
Do they have the right to request my resignation letter and do I need
to submit it?
That would depend on the law in Singapore. You don't need to submit it if you dont want to. I think you'd want to weigh in a lost opportunity versus sharing of the sacred resignation letter.