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

Yes.

The spelling, meaning, and grammatical function of "vermelho" are identical in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the vowels are typically more open and the syllables are pronounced more clearly. In Continental Portuguese, unstressed vowels are often "reduced" or more closed, meaning the "e" in the first syllable may sound much shorter, muffled, or nearly swallowed by a Brazilian ear.

Brazilian Portuguese

1. O carro é vermelho. (The car is red.)
2. Eu gosto de camiseta vermelha. (I like red t-shirts.)
3. O sinal ficou vermelho. (The traffic light turned red.)
4. O ônibus é vermelho. (The bus is red.)
5. O céu ficou vermelho no pôr do sol. (The sky turned red at sunset.)

Continental Portuguese

1. O carro é vermelho. (The car is red.)
2. Eu gosto de t-shirt vermelha. (I like red t-shirts.)
3. O semáforo ficou vermelho. (The traffic light turned red.)
4. O autocarro é vermelho. (The bus is red.)
5. O céu ficou vermelho ao pôr do sol. (The sky turned red at sunset.)