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

Yes.

The differences are strictly phonetic. In Brazilian Portuguese, the nasal diphthong "-éu" is typically pronounced with a more open vowel and a smoother, more fluid transition. In Continental Portuguese, the pronunciation is usually more closed and follows the distinct, stressed cadence and more pronounced nasalization characteristic of European Portuguese.

Brazilian Portuguese Examples

1. Não esqueça o seu chapéu, o sol está muito forte. (Don't forget your hat, the sun is very strong.)
2. Ele gosta de usar chapéu de palha na praia. (He likes to wear a straw hat at the beach.)
3. Esse chapéu combinou muito com a sua roupa. (That hat matched your outfit very well.)
4. O vento levou o meu chapéu para longe. (The wind blew my hat far away.)
5. Eu preciso comprar um chapéu novo para o verão. (I need to buy a new hat for the summer.)

Continental Portuguese Examples

1. Não te esqueças do teu chapéu, que o sol está forte. (Don't forget your hat, as the sun is strong.)
2. Ele gosta de usar chapéu de palha quando vai à praia. (He likes to use a straw hat when he goes to the beach.)
3. Esse chapéu fica-te muito bem. (That hat suits you very well.)
4. O vento levou-me o chapéu. (The wind took my hat away.)
5. Vou comprar um chapéu novo para o casamento. (I am going to buy a new hat for the wedding.)