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

Yes

The only difference between the two dialects regarding this word is pronunciation. In Brazilian Portuguese, the vowels are generally more "open" and clearly articulated, and the rhythm is more syllable-timed, meaning each syllable receives relatively equal weight. In Continental Portuguese, there is a much higher degree of vowel reduction; unstressed vowels (particularly the final 'a') are much more closed, shortened, or even nearly silent, resulting in a more clipped pronunciation.

Brazilian Portuguese

1. Vou limpar a mesa depois do jantar. (I am going to clean the table after dinner.)
2. Reserve uma mesa para quatro pessoas, por favor. (Reserve a table for four people, please.)
3. A mesa da sala de jantar é de madeira. (The dining room table is made of wood.)
4. Coloque o notebook em cima da mesa. (Put the notebook on the table.)
5. A mesa está posta para o café da manhã. (The table is set for breakfast.)

Continental Portuguese

1. Vou limpar a mesa depois do jantar. (I am going to clean the table after dinner.)
2. Reserve uma mesa para quatro pessoas, por favor. (Reserve a table for four people, please.)
3. A mesa da sala de jantar é de madeira. (The dining room table is made of wood.)
4. Põe o portátil em cima da mesa. (Put the laptop on the table.)
5. A mesa está posta para o pequeno-almoço. (The table is set for breakfast.)