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

Yes

The word "posso" is identical in meaning, spelling, and grammatical function (it is the first-person singular present indicative of the verb poder) in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the stressed vowel is often more open, and the unstressed "o" at the end is pronounced clearly as a "u" sound ([pɔsu]). In Continental Portuguese, the vowels are generally more closed, and the rhythm of speech is more compressed, which can make the unstressed syllables sound even more reduced.

Brazilian Portuguese examples:

1. Posso te ajudar com isso? (Can I help you with that?)
2. Eu não posso ir ao cinema hoje. (I can't go to the movies today.)
3. Posso pegar o seu celular um segundo? (Can I take your cell phone for a second?)
4. Você sabe se eu posso entrar? (Do you know if I can enter?)
5. Posso comer um pedaço de bolo? (Can I eat a piece of cake?)

Continental Portuguese examples:

1. Posso ajudar-te com isso? (Can I help you with that?)
2. Eu não posso ir ao cinema hoje. (I can't go to the movies today.)
3. Posso pegar no teu telemóvel um segundo? (Can I take your mobile phone for a second?)
4. Sabes se eu posso entrar? (Do you know if I can enter?)
5. Posso comer um pedaço de bolo? (Can I eat a piece of cake?)