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

Yes

The word "sombra" is identical in meaning, spelling, and grammar in both Brazilian and European Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the unstressed vowels are generally more open and clearly articulated. In European Portuguese, there is a much stronger tendency toward vowel reduction; therefore, the "o" in "sombra" is pronounced much more closed, often sounding like a very short, neutral "u".

Brazilian Portuguese

1. Vou descansar um pouco na sombra da árvore. (I'm going to rest for a bit in the shade of the tree.)
2. A sombra daquela nuvem passou bem rápido. (The shadow of that cloud passed very quickly.)
3. Ele gosta de ficar na sombra para não se queimar. (He likes to stay in the shade so he doesn't get burned.)
4. A sombra do prédio atrapalha o meu jardim. (The shadow of the building interferes with my garden.)
5. A sombra dele é muito grande. (His shadow is very large.)

European Portuguese

1. Estou a descansar um pouco à sombra da árvore. (I am resting for a bit in the shade of the tree.)
2. A sombra daquela nuvem passou muito depressa. (The shadow of that cloud passed very quickly.)
3. Ele gosta de ficar à sombra para não se queimar. (He likes to stay in the shade so he doesn't get burned.)
4. A sombra do prédio cai sobre o meu jardim. (The shadow of the building falls over my garden.)
5. A sombra dele é muito grande. (His shadow is very large.)