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

Yes

The only differences between the use of "caminhar" in Brazilian and Continental Portuguese are phonetic. In Brazilian Portuguese, the vowels are more open and unstressed vowels are clearly articulated. In Continental Portuguese, there is a significant reduction of unstressed vowels (vowel reduction), which makes the word sound much more "clipped" or compressed to a Brazilian ear. The spelling, grammar, and semantic meaning remain identical in both variants.

Brazilian Portuguese Usage

1. Eu gosto de caminhar no parque de manhã. (I like to walk in the park in the morning.)
2. Vamos caminhar um pouco para espairecer. (Let's walk a bit to clear our heads.)
3. Ele está caminhando todos os dias para emagrecer. (He is walking every day to lose weight.)
4. A situação da empresa começou a caminhar para uma solução. (The company's situation started to move towards a solution.)
5. É muito bom caminhar na praia ao pôr do sol. (It is very good to walk on the beach at sunset.)

Continental Portuguese Usage

1. Eu gosto de andar no parque de manhã. (I like to walk in the park in the morning.)
2. Vamos andar um pouco para distrair a cabeça. (Let's walk a bit to clear our heads.)
3. Ele anda todos os dias para perder peso. (He walks every day to lose weight.)
4. A situação da empresa começou a caminhar para uma solução. (The company's situation started to move towards a solution.)
5. É muito bom andar na praia ao pôr do sol. (It is very good to walk on the beach at sunset.)