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

Yes

The word "relógio" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. It refers to both a wristwatch and a wall clock. The only difference is pronunciation.

In Brazilian Portuguese, the vowels are more "open" and clearly articulated. The unstressed "o" at the end of the word is clearly pronounced as a /u/ sound. In Continental Portuguese, unstressed vowels are often "reduced" or swallowed, making the word sound much shorter and more clipped.

Brazilian Portuguese Usage

1. Eu perdi meu relógio no parque. (I lost my watch in the park.)
2. O relógio dele é muito caro. (His watch is very expensive.)
3. Você pode ajustar o relógio para mim? (Can you adjust the watch for me?)
4. Preciso de uma pilha nova para o relógio. (I need a new battery for the clock.)
5. Esse relógio digital é muito legal. (This digital watch is very cool.)

Continental Portuguese Usage

1. Perdi o meu relógio no parque. (I lost my watch in the park.)
2. O relógio dele é muito caro. (His watch is very expensive.)
3. Podes ajustar o relógio para mim? (Can you adjust the watch for me?)
4. Preciso de uma pilha nova para o relógio. (I need a new battery for the clock.)
5. Este relógio digital é muito giro. (This digital watch is very cool/nice.)