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

No

While the spelling and fundamental grammatical definition of "mim" are identical in both dialects, there are significant differences in typical usage.

1. Subject Usage: In many Brazilian dialects, "mim" is used colloquially as a subject of a verb (e.g., "para mim fazer"), which is considered a grammatical error in standard Portuguese. In Continental Portuguese, this usage is virtually non-existent; a speaker would always use the subject pronoun "eu".
2. Syntactic Preference: In Continental Portuguese, speakers much more frequently use the unstressed clitic pronoun "me" (e.g., "deu-me") to express the same idea that a Brazilian speaker would often express using the prepositional phrase "para mim" (e.g., "deu para mim").

Brazilian Portuguese

1. Isso é para mim? (Is this for me?)
2. Ele falou muito bem de mim. (He spoke very well of me.)
3. Você trouxe isso para mim? (Did you bring this to me?)
4. Você pode deixar isso para mim fazer? (Can you leave this for me to do? — Colloquial usage)
5. Ela guardou um doce para mim. (She saved a candy for me.)

European Portuguese

1. Isto é para mim? (Is this for me?)
2. Ele falou muito bem de mim. (He spoke very well of me.)
3. Trouxeste-me isto? (Did you bring this to me? — Using "me" instead of "para mim")
4. Deixa que eu faço isso. (Leave it, I will do this. — Using "eu" instead of "mim")
5. Ela guardou-me um doce. (She saved me a candy. — Using "me" instead of "para mim")