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

Yes

The word "desenhar" is identical in meaning, spelling, and conjugation in both Brazilian and Continental Portuguese. The differences between the two dialects are limited to pronunciation (Brazilian Portuguese features more open vowels, while Continental Portuguese features more closed or reduced vowels) and the grammatical construction used to express continuous actions (the gerund in Brazil versus the preposition "a" + infinitive in Portugal).

Brazilian Portuguese Examples

1. Eu adoro desenhar animais. (I love to draw animals.)
2. Ela está desenhando um retrato. (She is drawing a portrait.)
3. Você pode desenhar para mim? (Can you draw for me?)
4. Nós estamos desenhando no caderno. (We are drawing in the notebook.)
5. Ele desenhou um plano incrível. (He drew an incredible plan.)

Continental Portuguese Examples

1. Eu adoro desenhar animais. (I love to draw animals.)
2. Ela está a desenhar um retrato. (She is drawing a portrait.)
3. Podes desenhar para mim? (Can you draw for me?)
4. Nós estamos a desenhar no caderno. (We are drawing in the notebook.)
5. Ele desenhou-nos um plano incrível. (He drew an incredible plan for us.)