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

Yes.

The word "eu" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the pronunciation is generally more open and the diphthong is clearly articulated. In Continental Portuguese, the pronunciation tends to be more closed, and the vowels are often more reduced or "swallowed" depending on the speaker's regional accent.

Brazilian Portuguese

1. Eu estou no ônibus. (I am on the bus.)
2. Eu peguei o trem. (I caught the train.)
3. Eu gosto do meu cachorro. (I like my dog.)
4. Eu comi um doce. (I ate a sweet.)
5. Eu uso meu celular. (I use my cell phone.)

Continental Portuguese

1. Eu estou no autocarro. (I am on the bus.)
2. Eu apanhei o comboio. (I caught the train.)
3. Eu gosto do meu cão. (I like my dog.)
4. Eu comi um rebuçado. (I ate a sweet.)
5. Eu uso o meu telemóvel. (I use my mobile phone.)