No not exactly, the old woman is Mrs Busybody the tamagotchi matchmaker who appears if your tamagotchi has not got a friend or a sutiable partner. The tamagotchi you think she is trying to give you is a parter/lover who will give you tamababies. So if you want babies select "Yes" then fire works will go off and you will have babies, 24 hours later your 'main' tamagotchi character will go away and you will be left with the baby to start a generation all over again. If you select "No" she will go away and regualy keep coming (most days at 10:00am, 3:00pm or 7:00pm) until your tama is 16, when it has past 16 and you have chosen not to breed you will not be able to do so anymore and it'll turn into an old character (Male= Mr Busybody or Female= Mrs Busybody)
Good Luck
Ricky