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

No

While the spelling and the core meaning of "cozinhar" are identical in both dialects, there are differences in typical usage and pronunciation:

• Usage (Continuous Aspect): In Brazil, when expressing an action happening right now, the gerund is used (e.g., cozinhando). In Portugal, the construction "a + infinitive" is the standard (e.g., a cozinhar).
• Pronunciation: In Continental Portuguese, unstressed vowels are often reduced or "swallowed" (vowel reduction), making the word sound more closed. In Brazilian Portuguese, vowels are generally more open and pronounced clearly.

Brazilian Portuguese Examples

1. Eu estou cozinhando o jantar agora. (I am cooking dinner now.)
2. O que você está cozinhando para o almoço? (What are you cooking for lunch?)
3. Ela adora cozinhar para os amigos. (She loves to cook for friends.)
4. Nós vamos cozinhar algo especial hoje. (We are going to cook something special today.)
5. Você sabe cozinhar bem? (Do you know how to cook well?)

Continental Portuguese Examples

1. Eu estou a cozinhar o jantar agora. (I am cooking dinner now.)
2. O que é que estás a cozinhar para o almoço? (What are you cooking for lunch?)
3. Ela adora cozinhar para os amigos. (She loves to cook for friends.)
4. Nós vamos cozinhar algo especial hoje. (We are going to cook something special today.)
5. Tu sabes cozinhar bem? (Do you know how to cook well?)