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

Yes

The word "cama" is identical in meaning, spelling, and grammatical gender in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the unstressed final "a" is typically pronounced as a clear, open vowel [a]. In Continental Portuguese, unstressed vowels undergo "vowel reduction," meaning the final "a" is pronounced much more closed, sounding closer to a neutral "uh" [ɐ].

Brazilian Portuguese Examples

1. Eu vou para a cama agora. (I am going to bed now.)
2. A minha cama é muito macia. (My bed is very soft.)
3. Eu sempre arrumo a minha cama logo cedo. (I always make my bed early.)
4. Ele ficou na cama o dia inteirinho. (He stayed in bed the whole day.)
5. Essa cama é bem confortável. (This bed is quite comfortable.)

Continental Portuguese Examples

1. Vou para a cama agora. (I am going to bed now.)
2. A cama está bastante confortável. (The bed is quite comfortable.)
3. Eu faço a cama todos os dias. (I make the bed every day.)
4. Ele esteve na cama o dia todo. (He was in bed all day.)
5. Esta cama é demasiado grande para o quarto. (This bed is too big for the room.)