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

Yes

The word "trazem" is identical in spelling, meaning, and grammatical function in both Brazilian and European Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the unstressed "e" in the final syllable is more clearly articulated and audible. In European Portuguese, this vowel is highly reduced, often becoming nearly silent or sounding like a very short, muffled "i" or "u" sound.

Brazilian Portuguese

1. Eles trazem muita alegria para a nossa família. (They bring much joy to our family.)
2. As crianças trazem muita bagunça para a sala. (The children bring a lot of mess to the living room.)
3. Esses livros trazem muitos ensinamentos novos. (These books bring many new teachings.)
4. Eles trazem o lanche sempre na sexta-feira. (They always bring the snack on Friday.)
5. Os garçons trazem os pedidos rapidinho. (The waiters bring the orders very quickly.)

European Portuguese

1. Eles trazem muita alegria para a nossa família. (They bring much joy to our family.)
2. As crianças trazem muita confusão para a sala. (The children bring much chaos to the living room.)
3. Estes livros trazem muitos ensinamentos novos. (These books bring many new teachings.)
4. Eles trazem o lanche sempre à sexta-feira. (They always bring the snack on Friday.)
5. Os empregados trazem os pedidos rapidamente. (The employees bring the orders quickly.)