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

Yes

The word "trouxe" is identical in meaning, grammar, and spelling in both Brazilian and Continental Portuguese. It is the first-person singular of the verb trazer (to bring) in the pretérito perfeito do indicativo (simple past). The only difference lies in pronunciation:

• In Brazil, the final "e" is typically pronounced as a clear [i] sound (e.g., troo-see).
• In Portugal, the final "e" is highly reduced, often sounding like a very short, nearly silent neutral vowel, and the "x" is a distinct [ʃ] (sh) sound (e.g., troo-sh).

Brazilian Portuguese Examples

1. Eu trouxe um presente para você. (I brought a gift for you.)
2. Eu trouxe a comida que você pediu. (I brought the food you asked for.)
3. Eu trouxe meu cachorro para o parque. (I brought my dog to the park.)
4. Eu trouxe os documentos para a reunião. (I brought the documents for the meeting.)
5. Eu trouxe um pouco de café para a gente tomar. (I brought some coffee for us to drink.)

Continental Portuguese Examples

1. Eu trouxe um presente para ti. (I brought a gift for you.)
2. Eu trouxe a comida que pediste. (I brought the food you asked for.)
3. Eu trouxe o meu cão para o parque. (I brought my dog to the park.)
4. Eu trouxe os documentos para a reunião. (I brought the documents for the meeting.)
5. Eu trouxe um pouco de café para tomarmos. (I brought some coffee for us to drink.)