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

Yes.

The spelling, meaning, and grammatical gender of "mar" are identical in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Continental Portuguese, the "r" is typically pronounced as a uvular or guttural sound (similar to the French "r"), whereas in Brazilian Portuguese, the pronunciation of the "r" varies by region but is commonly an alveolar tap or a retroflex sound (similar to the American English "r" in "car").

Brazilian Portuguese

1. O mar está muito calmo hoje. (The sea is very calm today.)
2. Eu adoro nadar no mar. (I love swimming in the sea.)
3. A gente vai passar o dia no mar. (We are going to spend the day at the sea.)
4. Tem muita onda no mar hoje. (There are many waves in the sea today.)
5. O mar tá muito bonito. (The sea is very beautiful.)

Portuguese (Portugal)

1. O mar está muito calmo hoje. (The sea is very calm today.)
2. Eu adoro nadar no mar. (I love swimming in the sea.)
3. Nós vamos passar o dia no mar. (We are going to spend the day at the sea.)
4. Há muitas ondas no mar hoje. (There are many waves in the sea today.)
5. O mar está muito bonito. (The sea is very beautiful.)