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

No.

While the fundamental meanings of acabar (to finish, to end, to run out) are identical in both dialects, there is a significant difference in grammatical usage regarding the expression "to end up [doing something]."

In Brazilian Portuguese, this is typically expressed using the gerund (e.g., acabou saindo). In Continental Portuguese, the standard construction uses the preposition por followed by the infinitive (e.g., acabou por sair). Additionally, Brazilians frequently use the colloquial construction "acabar que" to mean "it turned out that," a structure rarely used in Portugal.

Brazilian Portuguese

1. Eu acabei de comer. (I just finished eating.)
2. O açúcar acabou. (The sugar ran out.)
3. Ele acabou ficando bravo. (He ended up getting angry.)
4. Ele acabou perdendo o ônibus. (He ended up missing the bus.)
5. O filme acabou muito tarde. (The movie ended very late.)

European Portuguese

1. Eu acabei de comer. (I just finished eating.)
2. O açúcar acabou. (The sugar ran out.)
3. Ele acabou por ficar bravo. (He ended up getting angry.)
4. Ele acabou por perder o autocarro. (He ended up missing the bus.)
5. O filme acabou muito tarde. (The movie ended very late.)