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

No

While the core meaning of "levar" remains the same in both dialects, the answer is No because there are significant differences in typical usage and grammatical construction surrounding the verb. In Brazilian Portuguese, it is common in informal speech to use subject pronouns as direct objects (e.g., levar ele instead of the grammatically formal levá-lo) and to use the preposition em (e.g., levar no shopping). In Portugal, the standard usage relies on clitic pronouns (e.g., levá-lo) and the preposition a (e.g., levar ao shopping).

Brazilian Portuguese Usage

1. Eu vou levar ele na escola. (I am going to take him to school.)
2. Você vai levar o cachorro para passear? (Are you going to take the dog for a walk?)
3. Isso vai levar muito tempo para terminar. (This is going to take a lot of time to finish.)
4. Leva o casaco porque vai esfriar. (Take the jacket because it's going to get cold.)
5. Ele levou um fora daquela menina. (He got rejected by that girl.)

Continental Portuguese Usage

1. Eu vou levá-lo à escola. (I am going to take him to school.)
2. Vais levar o cão para passear? (Are you going to take the dog for a walk?)
3. Isto vai levar muito tempo a terminar. (This is going to take a lot of time to finish.)
4. Leva o casaco porque vai arrefecer. (Take the jacket because it's going to get cold.)
5. Ele levou um fora daquela rapariga. (He got rejected by that girl.)