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

No

While the fundamental meaning of "strolling" or "walking for pleasure" is identical, there are differences in typical usage and pronunciation. In Brazilian Portuguese, "passear" is used more broadly to mean "to travel," "to visit," or "to go on a trip" for leisure (e.g., "vou passear em Gramado"). In Portugal, the word is more strictly associated with a short walk or a stroll; for longer trips, a Portuguese person would almost exclusively use "viajar." Additionally, the pronunciation differs significantly: Brazilian Portuguese features more open, stable vowels, whereas Continental Portuguese features heavy vowel reduction, where unstressed vowels often become much shorter or nearly silent.

Brazilian Portuguese Examples

1. Eu gosto de passear no parque. (I like to walk in the park.)
2. Vamos passear com o cachorro? (Shall we walk the dog?)
3. Nós passeamos no shopping ontem. (We strolled around the mall yesterday.)
4. Ela gosta de passear de bicicleta à tarde. (She likes to ride her bike for pleasure in the afternoon.)
5. Vou passear em Minas Gerais nas férias. (I am going to travel/visit Minas Gerais during the holidays.)

Continental Portuguese Examples

1. Eu gosto de dar um passeio no parque. (I like to go for a walk in the park.)
2. Vamos passear com o cão? (Shall we walk the dog?)
3. Nós demos uma volta no centro ontem. (We took a stroll around the center yesterday.)
4. Ela gosta de passear de bicicleta à tarde. (She likes to ride her bike for pleasure in the afternoon.)
5. Vou viajar para Minas Gerais nas férias. (I am going to travel to Minas Gerais during the holidays.)