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

No.

While the meaning, spelling, and core grammar of "mudar" are identical in both dialects, there is a difference in typical usage and pronunciation:

• Typical Usage: When referring to changing one's residence, Brazilians typically use the prepositional form "mudar de" (e.g., mudar de casa), whereas Portuguese people more naturally use the reflexive form "mudar-se" (e.g., mudar-me de casa).
• Pronunciation: In Brazilian Portuguese, vowels are generally more open and the "r" is often more pronounced or retroflex. In Continental Portuguese, vowels are often "reduced" (more closed or almost silent), and the "r" is typically more alveolar.

Brazilian Portuguese Examples:

1. Eu vou mudar de casa amanhã. (I am going to move house tomorrow.)
2. Você mudou de ideia sobre o trabalho. (You changed your mind about the job.)
3. A gente precisa mudar esse plano. (We need to change this plan.)
4. O tempo mudou muito rápido hoje. (The weather changed very quickly today.)
5. Eu quero mudar meu visual. (I want to change my look.)

Portuguese (Portugal) Examples:

1. Eu vou mudar-me de casa amanhã. (I am going to move house tomorrow.)
2. Tu mudaste de ideia sobre o trabalho. (You changed your mind about the job.)
3. Nós precisamos de mudar este plano. (We need to change this plan.)
4. O tempo mudou muito depressa hoje. (The weather changed very quickly today.)
5. Eu quero mudar o meu visual. (I want to change my look.)