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

Yes

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

• Brazilian Portuguese: The pronunciation is generally more "open" and musical. The vowels are clearly articulated, and the "ei" diphthong is pronounced with a clear, open sound (/ka'dej.ɾɐ/).
• Continental Portuguese: The pronunciation is more "closed" and stress-timed. Unstressed vowels are often reduced or almost swallowed, making the word sound more clipped or staccato (/kɐ'dej.ɾɐ/).

Brazilian Portuguese Examples

1. Senta nessa cadeira aqui. (Sit in this chair here.)
2. A cadeira da sala está quebrada. (The living room chair is broken.)
3. Eu vou comprar uma cadeira nova para o escritório. (I am going to buy a new chair for the office.)
4. Essa cadeira é muito confortável, né? (This chair is very comfortable, right?)
5. Pode deixar a cadeira no canto. (You can leave the chair in the corner.)

Continental Portuguese Examples

1. Senta-te nesta cadeira. (Sit in this chair.)
2. A cadeira da sala está partida. (The living room chair is broken.)
3. Vou comprar uma cadeira nova para o escritório. (I am going to buy a new chair for the office.)
4. Esta cadeira é bastante confortável. (This chair is quite comfortable.)
5. Podes pôr a cadeira no canto. (You can put the chair in the corner.)