Is the word cozinho the same in Brazilian and European Portuguese?

Yes

The word "cozinho" is the first-person singular present indicative of the verb cozinhar (to cook). There are no differences in meaning, grammar, or spelling between Brazilian and Continental Portuguese. The only distinction lies in pronunciation: in Brazil, vowels are generally more open and clearly articulated, whereas in Portugal, unstressed vowels (such as the "o" in the first syllable) tend to be more closed or even reduced.

Brazilian Portuguese Examples:

1. Eu cozinho o almoço para as crianças. (I cook lunch for the children.)
2. Eu cozinho um macarrão bem gostoso. (I cook a very tasty pasta.)
3. Eu cozinho o meu jantar sempre à noite. (I cook my dinner always at night.)
4. Eu cozinho o ovo cozido. (I cook the boiled egg.)
5. Eu cozinho o que tem na geladeira. (I cook what is in the fridge.)

Portuguese (Portugal) Examples:

1. Eu cozinho o almoço para as crianças. (I cook lunch for the children.)
2. Eu cozinho uma massa muito saborosa. (I cook a very tasty pasta.)
3. Eu cozinho o meu jantar sempre à noite. (I cook my dinner always at night.)
4. Eu cozinho o ovo cozido. (I cook the boiled egg.)
5. Eu cozinho o que está no frigorífico. (I cook what is in the fridge.)