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

Yes

The word "conseguir" is identical in meaning, grammar, spelling, and typical usage in both Brazilian and Continental Portuguese. The only differences lie in pronunciation. In Continental Portuguese, the vowels are generally more closed or "reduced" (the "e" in conseguir may sound more like a very short, almost silent "i"), and the "s" in certain positions (such as at the end of a syllable) is pronounced as a "sh" sound ([ʃ]), whereas in most of Brazil, it is pronounced as a "s" sound ([s]).

Brazilian Portuguese Examples:

1. Eu não consegui dormir nada ontem. (I couldn't sleep at all yesterday.)
2. Você consegue me ajudar com isso? (Can you help me with this?)
3. A gente conseguiu comprar os ingressos. (We managed to buy the tickets.)
4. Ele conseguiu o que queria. (He got what he wanted.)
5. Não consegui achar o seu número. (I couldn't find your number.)

European Portuguese Examples:

1. Eu não consegui dormir nada ontem. (I couldn't sleep at all yesterday.)
2. Tu consegues ajudar-me com isto? (Can you help me with this?)
3. Nós conseguimos comprar os bilhetes. (We managed to buy the tickets.)
4. Ele conseguiu o que pretendia. (He got what he intended.)
5. Não consegui encontrar o teu número. (I couldn't find your number.)