The parents of the children would receive the money or we would very quickly have a low or even negative birthrate. However, at the same time the amount per child would need to go down after the first one or two or the population would explode.
Would you argue that if a man decides he doesn't want to pay for a child he helped conceive he shouldn't be forced to?
This is why it isn't a simple question. If it is ok to absolve the woman involved of all responsibility then it follows that it must also be ok to absolve the man as well.
The difference between a company losing your information and the government losing it is very simple. You can sue the company and the government will enforce the ruling should you win. If you attempt to sue the government not only can it decline to accept the suit in the first place but it can also choose how or even if it will enforce the ruling should you win.