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

Yes

The word "chão" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only differences lie in pronunciation. In Brazilian Portuguese, the nasal vowel "ão" tends to be pronounced with more openness and a more melodic, drawn-out nasalization. In Continental Portuguese, the pronunciation is often more "closed" or attenuated, and the rhythm of the word follows the more stress-timed nature of European Portuguese, making the nasal sound slightly more subtle.

Brazilian Portuguese Examples

1. O cachorro está dormindo no chão. (The dog is sleeping on the floor.)
2. Eu derrubei água no chão. (I spilled water on the floor.)
3. O chão da sala está brilhando. (The living room floor is shining.)
4. Não deixe seus brinquedos no chão. (Don't leave your toys on the floor.)
5. Ele caiu no chão e se machucou. (He fell on the ground and hurt himself.)

Continental Portuguese Examples

1. O cão está a dormir no chão. (The dog is sleeping on the floor.)
2. Derramei água no chão. (I spilled water on the floor.)
3. O chão da sala está a brilhar. (The living room floor is shining.)
4. Não deixes os teus brinquedos no chão. (Don't leave your toys on the floor.)
5. Ele caiu no chão e magoou-se. (He fell on the ground and hurt himself.)