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

Yes

The spelling, meaning, and grammatical function of "exame" are identical in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazil, the final "e" is typically pronounced as a clear [i] (like the "ee" in "see"). In Portugal, the final "e" is a reduced, closed vowel [ɨ], which sounds more muffled or like a very short, neutralized sound.

Brazilian Portuguese Examples

1. Eu tenho um exame de sangue amanhã. (I have a blood test tomorrow.)
2. O aluno está muito nervoso para o exame. (The student is very nervous for the exam.)
3. O médico pediu um exame de raio-X. (The doctor requested an X-ray exam.)
4. O exame de admissão foi cancelado. (The entrance exam was canceled.)
5. Você já fez o seu exame de vista? (Have you had your eye exam yet?)

Portuguese (Portugal) Examples

1. Tenho de fazer as análises ao sangue amanhã. (I have to do blood tests tomorrow.)
2. O aluno está muito nervoso para o exame. (The student is very nervous for the exam.)
3. O médico pediu um raio-X. (The doctor requested an X-ray.)
4. O exame de admissão foi cancelado. (The entrance exam was canceled.)
5. Já fiz o meu exame de vista. (I have already had my eye exam.)