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

No

The difference lies in the typical usage and meaning of the word.

In Brazilian Portuguese, "academia" is the standard, everyday word for a "gym" or fitness center. While Brazilians do use the word to refer to the "academic world" (universities, research, and scholars), this is reserved for formal or intellectual contexts. In daily conversation, if a Brazilian says "Eu vou à academia," they almost certainly mean they are going to exercise.

In Continental Portuguese, "academia" refers almost exclusively to the "academic sphere" (the community of scholars, universities, and literature). To refer to a place where one goes to lift weights or exercise, a Portuguese person uses the word ginásio.

Brazilian Portuguese Examples (using academia as "gym")

1. Eu vou à academia todos os dias. (I go to the gym every day.)
2. A academia está muito lotada hoje. (The gym is very crowded today.)
3. Quanto custa a mensalidade da academia? (How much is the gym membership?)
4. Ele treina na academia logo cedo. (He trains at the gym early in the morning.)
5. Preciso comprar uma roupa nova para a academia. (I need to buy new clothes for the gym.)

Portuguese Examples (using ginásio to express the same idea)

1. Eu vou ao ginásio todos os dias. (I go to the gym every day.)
2. O ginásio está muito cheio hoje. (The gym is very crowded today.)
3. Quanto custa a mensalidade do ginásio? (How much is the gym membership?)
4. Ele treina no ginásio logo cedo. (He trains at the gym early in the morning.)
5. Preciso de sapatilhas novas para o ginásio. (I need new sneakers for the gym.)