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

No

The difference lies in regência verbal (verb government/syntax). While the core meaning of the word remains the same, the grammatical usage differs significantly between the two dialects. In Brazilian Portuguese, especially in colloquial speech, assistir is frequently used as a transitive direct verb, meaning the object follows the verb without a preposition (e.g., assisto o filme). In Continental Portuguese, the verb is strictly treated as a transitive indirect verb, requiring the preposition a (e.g., assisto ao filme). Additionally, vocabulary for objects often differs, such as "cell phone" being celular in Brazil and telemóvel in Portugal.

Brazilian Portuguese usage:

1. Eu assisto o jogo toda semana. (I watch the game every week.)
2. Eu assisto os filmes de terror. (I watch horror movies.)
3. Eu assisto o desenho com as crianças. (I watch the cartoon with the children.)
4. Eu assisto a novela no intervalo. (I watch the soap opera during the break.)
5. Eu assisto o vídeo no celular. (I watch the video on the cell phone.)

Continental Portuguese usage:

1. Eu assisto ao jogo todas as semanas. (I watch the game every week.)
2. Eu assisto aos filmes de terror. (I watch the horror movies.)
3. Eu assisto ao desenho animado com as crianças. (I watch the cartoon with the children.)
4. Eu assisto à novela no intervalo. (I watch the soap opera during the break.)
5. Eu assisto ao vídeo no telemóvel. (I watch the video on the mobile phone.)