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

Yes

The word "nesse" is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. It is the contraction of the preposition em (in) and the demonstrative pronoun esse (that). The only significant difference is in pronunciation. In Brazilian Portuguese, the unstressed final "e" is typically pronounced as a high, front vowel [i], making the word sound like "nessi." In Continental Portuguese, the unstressed "e" is much more heavily reduced, often sounding like a schwa [ə] or becoming almost silent in rapid speech.

Brazilian Portuguese Examples

1. Eu vi o problema nesse relatório. (I saw the problem in that report.)
2. Nesse caso, não podemos fazer nada. (In that case, we can't do anything.)
3. O que você colocou nesse copo? (What did you put in that glass?)
4. Nesse lugar, sempre chove muito. (In that place, it always rains a lot.)
5. Não acredito nesse seu argumento. (I don't believe in that argument of yours.)

Continental Portuguese Examples

1. Eu vi o problema nesse relatório. (I saw the problem in that report.)
2. Nesse caso, não podemos fazer nada. (In that case, we can't do anything.)
3. O que é que puseste nesse copo? (What is it that you put in that glass?)
4. Nesse sítio, chove sempre muito. (In that place, it always rains a lot.)
5. Não acredito nesse teu argumento. (I don't believe in that argument of yours.)