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

No

While the fundamental meaning of "trazer" (to bring) is identical in both dialects, there are significant differences in grammar (specifically syntax and clitic placement) and usage (preposition preference). There are no differences in spelling.

• Grammar (Clitic Placement): In Brazilian Portuguese (BP), it is extremely common and natural to place object pronouns before the verb (proclisis), such as "me trazer." In Continental Portuguese (CP), the standard and natural usage is to place the pronoun after the verb (enclisis), such as "trazer-me."
• Usage (Prepositions): Brazilians heavily favor the preposition "para" to indicate destination or recipient (e.g., "trazer para mim"). Portuguese speakers more frequently use the preposition "a" (e.g., "trazer-me a").
• Pronunciation: Brazilian Portuguese is generally more "open" and melodic, with clearly articulated vowels. Continental Portuguese is "closed" and stress-timed, often reducing or swallowing unstressed vowels, which can make the word sound more consonant-heavy.
• Vocabulary context: While "trazer" remains the same, the objects being brought often use different words (e.g., celular in Brazil vs. telemóvel in Portugal).

Brazilian Portuguese Examples

1. Você pode me trazer um copo d'água? (Can you bring me a glass of water?)
2. Eu vou trazer o controle da TV. (I'm going to bring the TV remote.)
3. Ela trouxe o lanche para a escola. (She brought the snack to the school.)
4. Não esquece de trazer o seu celular. (Don't forget to bring your cell phone.)
5. Me traz aquele livro ali? (Bring me that book over there?)

Continental Portuguese Examples

1. Podes trazer-me um copo de água? (Can you bring me a glass of water?)
2. Traze-me o comando da televisão. (Bring me the TV remote.) Note: "Comando" is used in Portugal instead of "controle."
3. O rapaz trouxe o lanche à escola. (The boy brought the snack to the school.) Note: Uses "à" instead of "para a."
4. Não te esqueças de trazer o teu telemóvel. (Don't forget to bring your mobile phone.) Note: "Telemóvel" is used in Portugal instead of "celular."
5. Tragam-me os documentos, por favor. (Bring me the documents, please.)