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

Yes.

The word "começar" is identical in meaning, spelling, and grammatical conjugation in both Brazilian and Continental Portuguese. The differences between the two dialects are strictly phonetic:

• Vowels: Brazilian Portuguese tends to have more "open" and melodic vowels. In Continental Portuguese, vowels are more "closed" or reduced, and unstressed vowels are often "swallowed" or shortened.
• The terminal "r": In many Brazilian dialects, the final "r" in "começar" is pronounced as a soft, aspirated sound (similar to the English "h") or is almost silent. In Portugal, the "r" is typically more pronounced as a guttural or uvular sound.

Brazilian Portuguese Examples:

1. Eu vou começar o curso amanhã. (I am going to start the course tomorrow.)
2. A aula começa às oito horas. (The class starts at eight o'clock.)
3. Eles começaram a estudar muito. (They started studying a lot.)
4. Não comece essa conversa agora. (Don't start that conversation now.)
5. Vamos começar o filme? (Shall we start the movie?)

Continental Portuguese Examples:

1. Vou começar o livro hoje. (I am going to start the book today.)
2. O jantar começa às nove. (Dinner starts at nine.)
3. Eles começaram o projeto. (They started the project.)
4. Não comece esse trabalho. (Don't start that work.)
5. Vamos começar a reunião? (Shall we start the meeting?)