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

No.

The difference lies in usage and syntax, specifically regarding clitic placement (where the pronoun is positioned in relation to the verb).

In Brazilian Portuguese, there is a strong preference for proclisis (placing the pronoun before the verb), even in informal speech. It is very common, and widely accepted in colloquial Brazil, to start a sentence with the word "me." In Continental Portuguese, the standard is enclisis (placing the pronoun after the verb). In Portugal, starting a sentence with an object pronoun like "me" is considered grammatically incorrect.

Brazilian Portuguese Examples

1. Me empresta uma caneta? (Can you lend me a pen?)
2. Ele me ligou ontem. (He called me yesterday.)
3. Me avisa quando você chegar. (Let me know when you arrive.)
4. Você me viu na festa. (You saw me at the party.)
5. Me traz um café, por favor. (Bring me a coffee, please.)

Continental Portuguese Examples

1. Empresta-me uma caneta? (Can you lend me a pen?)
2. Ele ligou-me ontem. (He called me yesterday.)
3. Avisa-me quando chegares. (Let me know when you arrive.)
4. Viste-me na festa? (Did you see me at the party?)
5. Traz-me um café, por favor. (Bring me a coffee, please.)