No, they don't, A. 'manner', with no 's', means pretty much the same as 'way' but 'manners' with an 's' has a different meaning and so it could not be used in this situation.
Thanks, riverkid, for the clear explanation.
I've always confused manner with manners.
For the sake of clarity, are the following right?
Alice is a lady because she has good manners.
Mind your manners when eating.