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

Yes

The word "cozinha" is identical in spelling, meaning, and grammar in both Brazilian and European Portuguese. The difference is purely phonetic. In Brazilian Portuguese, the vowels are generally more "open" and the pronunciation is more melodic and rhythmic. In European Portuguese, unstressed vowels are often more "closed" or even reduced (almost swallowed), which can make the word sound more clipped or subtle to a Brazilian ear.

Brazilian Portuguese

1. A cozinha está uma bagunça. (The kitchen is a mess.)
2. Eu adoro cozinhar na minha cozinha nova. (I love cooking in my new kitchen.)
3. A cozinha desse restaurante é incrível. (The kitchen of that restaurant is incredible.)
4. Ele está na cozinha preparando o almoço. (He is in the kitchen preparing lunch.)
5. Nossa cozinha é muito pequena. (Our kitchen is very small.)

Portuguese (Portugal)

1. A cozinha está uma confusão. (The kitchen is a mess.)
2. Adoro cozinhar na minha cozinha nova. (I love cooking in my new kitchen.)
3. A cozinha deste restaurante é incrível. (The kitchen of this restaurant is incredible.)
4. Ele está na cozinha a preparar o almoço. (He is in the kitchen preparing lunch.)
5. A nossa cozinha é bastante pequena. (Our kitchen is quite small.)