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

Yes

The word "pai" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the diphthong "ai" is generally pronounced with a more open "a" and a clear, elongated "i." In Continental Portuguese, the vowels are often more closed or reduced, and the rhythm of the word follows the more "staccato" or syllable-timed nature of European Portuguese.

Brazilian Portuguese Examples

1. Meu pai trabalha muito. (My father works a lot.)
2. Eu amo meu pai. (I love my-father.)
3. Meu pai me deu um presente. (My father gave me a gift.)
4. O meu pai viajou ontem. (My father traveled yesterday.)
5. Você falou com o seu pai? (Did you speak to your father?)

Continental Portuguese Examples

1. O meu pai trabalha muito. (My father works a lot.)
2. Eu amo o meu pai. (I love my father.)
3. O meu pai deu-me um presente. (My father gave me a gift.)
4. O meu pai viajou ontem. (My father traveled yesterday.)
5. Viste o teu pai? (Did you see your father?)