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

Yes

The word "desenham" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese; it is the third-person plural present indicative of the verb desenhar (to draw). The only difference is pronunciation. In Brazil, the vowels tend to be more open and the nasalization of the final "m" is more pronounced. In Portugal, unstressed vowels (like the "e" in the first syllable) are often more closed or even reduced/nearly silent, and the phonetic rhythm is more stress-timed.

Brazilian Portuguese Examples:

1. As crianças desenham muito bem. (The children draw very well.)
2. Eles desenum paisagens no caderno. (They draw landscapes in the notebook.)
3. Os alunos desenham o mapa da sala. (The students draw the map of the classroom.)
4. Eles desenham o que veem na tela. (They draw what they see on the screen.)
5. Os artistas desenham murais coloridos. (The artists draw colorful murals.)

Continental Portuguese Examples:

1. Os miúdos desenham muito bem. (The kids draw very well.)
2. Eles desenham paisagens no caderno. (They draw landscapes in the notebook.)
3. Os alunos desenham o mapa da sala. (The students draw the map of the classroom.)
4. Eles desenham o que veem no ecrã. (They draw what they see on the screen.)
5. Os artistas desenham murais coloridos. (The artists draw colorful murals.)