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

Yes

The word "braço" is identical in meaning, grammar, and spelling in both Brazilian and Continental Portuguese. The differences are strictly phonetic. In Brazilian Portuguese, the "r" is often realized as a retroflex [ɻ] (in many regions) or a simple alveolar tap, and the unstressed final "o" is pronounced as a clear [u]. In Continental Portuguese, the "r" is typically an alveolar tap or trill, and the unstressed "o" is more heavily reduced and closed, often sounding like a very short, almost swallowed [u].

Brazilian Portuguese

1. Eu quebrei o braço jogando futebol. (I broke my arm playing soccer.)
2. Ele é meu braço direito na empresa. (He is my right hand in the company.)
3. O braço desta cadeira está quebrado. (The arm of this chair is broken.)
4. Precisamos de mais braços para carregar essa caixa. (We need more hands/people to carry this box.)
5. Estou com muita dor no braço. (I have a lot of pain in my arm.)

Continental Portuguese

1. Eu parti o braço a jogar futebol. (I broke my arm playing soccer.)
2. Ele é o meu braço direito na empresa. (He is my right hand in the company.)
3. O braço deste sofá está partido. (The arm of this sofa is broken.)
4. Precisamos de mais braços para a colheita. (We need more hands/people for the harvest.)
5. Estou com o braço muito dorido. (My arm is very sore.)