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

No.

While the fundamental meaning of the word remains the same, there are significant differences in grammar and syntax between the two dialects:

• Clitic Placement (Syntax): In Brazilian Portuguese, it is standard and natural to use proclisis, placing the object pronoun before the verb (e.g., me trata). In Continental Portuguese, enclisis is the norm, placing the pronoun after the verb (e.g., trata-me).
• Verb Aspect (Grammar): To express an ongoing action, Brazilians use the gerund (tratando), whereas the Portuguese typically use the construction "a + infinitive" (a tratar).
• Usage/Vocabulary: While the word "tratar" is used identically, the accompanying vocabulary often shifts (e.g., cachorro in Brazil vs. cão in Portugal).

Brazilian Portuguese

1. Ele me trata muito bem. (He treats me very well.)
2. Eu vou tratar desse problema agora. (I am going to deal with this problem now.)
3. Ela está tratando o cachorro com muito carinho. (She is treating/caring for the dog with a lot of affection.)
4. Você precisa tratar de todos os documentos. (You need to handle all the documents.)
5. Isso não se trata de algo fácil. (This is not about something easy.)

Continental Portuguese

1. Ele trata-me muito bem. (He treats me very well.)
2. Eu vou tratar desse problema agora. (I am going to deal with this problem now.)
3. Ela está a tratar o cão com muito carinho. (She is treating/caring for the dog with a lot of affection.)
4. Tu precisas de tratar de todos os documentos. (You need to handle all the documents.)
5. Isto não se trata de algo fácil. (This is not about something easy.)