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

Yes. The word "caminham" is identical in meaning, grammar, and spelling in both Brazilian and Continental Portuguese. The only difference is in pronunciation.

In Brazilian Portuguese, the unstressed vowel "i" is typically pronounced clearly and openly ([ka.mi.ˈɲɐ̃w]). In Continental Portuguese, unstressed vowels undergo significant reduction; the "i" is often shortened or "swallowed," making the word sound much closer to [kɐ.mi.ˈɲɐ̃w].

Brazilian Portuguese Examples

1. Eles caminham no parque todas as manhãs. (They walk in the park every morning.)
2. As crianças caminham juntas para a escola. (The children walk together to school.)
3. Muitos turistas caminham pela orla do Rio. (Many tourists walk along the Rio waterfront.)
4. Elas caminham devagar para apreciar a vista. (They walk slowly to enjoy the view.)
5. As pessoas caminham pela calçada com cuidado. (People walk on the sidewalk carefully.)

Continental Portuguese Examples

1. Eles andam no parque todas as manhãs. (They walk in the park every morning. — A Portuguese person would more naturally use "andam" for simple everyday walking.)
2. As crianças caminham juntas para a escola. (The children walk together to school.)
3. Muitos turistas caminham pela marginal. (Many tourists walk along the coastal road. — The word "marginal" is more common in Portugal for a waterfront/coastal road.)
4. Elas andam devagar para apreciar a vista. (They walk slowly to enjoy the view.)
5. As pessoas andam pelo passeio com cuidado. (People walk on the sidewalk carefully. — The word "passeio" is often used for sidewalk in Portugal.)