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

Yes

The word "nome" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only difference is pronunciation:

• Brazilian Portuguese: The final unstressed "e" is typically pronounced as a high, closed "i" (sounding like nō-mee).
• Continental Portuguese: The final "e" is a "reduced" vowel, meaning it is much more closed, shorter, and almost sounds like a muffled "uh" or even disappears (sounding closer to nōm-uh).

Brazilian Portuguese Examples

1. Qual é o seu nome? (What is your name?)
2. Eu esqueci o nome dele. (I forgot his name.)
3. Escreva seu nome aqui. (Write your name here.)
4. O nome do cachorro é Bob. (The dog's name is Bob.)
5. Você sabe o nome dessa rua? (Do you know the name of this street?)

Continental Portuguese Examples

1. Como se chama? (What is your name?)
2. Não me lembro do nome dele. (I don't remember his name.)
3. Escreva o seu nome aqui. (Write your name here.)
4. O nome do cão é Bob. (The dog's name is Bob.)
5. Sabes o nome desta rua? (Do you know the name of this street?)