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

Yes

The meaning, spelling, and grammatical function of the word "nomes" (the plural of nome) are identical in both Brazilian and Continental Portuguese. The only difference is in pronunciation.

In Brazilian Portuguese, vowels are generally more open and clearly articulated, so the "e" in "nomes" is pronounced clearly. In Continental Portuguese, unstressed vowels tend to be "reduced" or swallowed, meaning the "e" sounds much shorter, more closed, or almost disappears, making the word sound more clipped.

Brazilian Portuguese Examples

1. Escrevi os nomes na lista. (I wrote the names on the list.)
2. Você sabe os nomes deles? (Do you know their names?)
3. Eu não lembro os nomes das crianças. (I don't remember the children's names.)
4. Os nomes das ruas são difíceis. (The names of the streets are difficult.)
5. Escreva os seus nomes aqui. (Write your names here.)

Continental Portuguese Examples

1. Escrevi os nomes na lista. (I wrote the names on the list.)
2. Tu sabes os nomes deles? (Do you know their names?)
3. Eu não me lembro dos nomes das crianças. (I don't remember the children's names.)
4. Os nomes das ruas são difíceis. (The names of the streets are difficult.)
5. Escrevam os vossos nomes aqui. (Write your names here.)