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

Yes

The word "aula" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels are generally more "open" and the final "a" is clearly and distinctly articulated. In Continental Portuguese, unstressed vowels (like the final "a") tend to be "reduced" or "closed," making the word sound shorter and more clipped.

Brazilian Portuguese examples:

1. A aula de matemática foi muito legal. (Math class was very cool.)
2. Eu vou chegar atrasado para a aula. (I am going to be late for class.)
3. Você viu o que o professor passou na aula? (Did you see what the teacher covered in class?)
4. A aula de natação é de manhã. (Swimming class is in the morning.)
5. Eu preciso estudar para a aula de amanhã. (I need to study for tomorrow's class.)

Portuguese (Continental) examples:

1. A aula de matemática foi muito gira. (Math class was very cool.)
2. Vou chegar atrasado à aula. (I am going to be late to the class.)
3. Viste o que o professor explicou na aula? (Did you see what the teacher explained in class?)
4. A aula de natação decorre de manhã. (Swimming class takes place in the morning.)
5. Tenho de estudar para a aula de amanhã. (I have to study for tomorrow's class.)