I believe you wanted to compare "twoje" and "wasze", because "nasze" means "our", so it's definitely a different thing.
So "twoje" is singular 'your', and "wasze" is plural 'your'. So basically it's just a matter of who you are talking to/referring to.
For example even if you're talking to your friend John, you may say "twoje dziecko", having just him in mind, or you may say "wasze dziecko", treating it like "your wife's and yours child". If you're talking to both parents, you really should say "wasze dziecko".
"drinks" is also accepted. However, if you make some error in your translation, the 'correct' answer you'll be shown will probably be the "is drinking" version.
"Czy dziecko mleko pije ? ie does he drink milk" is a different sentence and wouldn't be accepted here.