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

Yes

The word "mesmo" is identical in meaning, grammar, and spelling in both Brazilian and Continental Portuguese. It can function as an adjective (meaning "same"), an adverb (meaning "really," "truly," or "even"), or a pronoun (meaning "myself," "himself," etc.). The only difference is pronunciation: in Brazilian Portuguese, the vowels are generally more open and the "e" is pronounced clearly, whereas in Continental Portuguese, the vowels are more closed, reduced, or often "swallowed" in unstressed positions.

Brazilian Portuguese usage

1. Eu quero o mesmo que você. (I want the same as you.)
2. É mesmo isso? (Is that really it?)
3. Eu mesmo vou fazer o bolo. (I will make the cake myself.)
4. Ele saiu agora mesmo. (He left just now.)
5. Mesmo que chova, eu vou. (Even if it rains, I will go.)

Continental Portuguese usage

1. Eu quero o mesmo que tu. (I want the same as you.)
2. É verdade? (Is it true?)
3. Eu próprio vou fazer o bolo. (I will make the cake myself.)
4. Ele saiu agora mesmo. (He left just now.)
5. Ainda que chova, eu vou. (Even if it rains, I will go.)