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

No

While the core meaning of "lado" is identical in both dialects, there are differences in typical usage and pronunciation. In Brazilian Portuguese, it is extremely common to use the prepositional phrase "do lado de" to indicate proximity (meaning "next to"). In Continental Portuguese, the standard and more natural usage is "ao lado de." Additionally, the pronunciation of the vowels differs significantly; Brazilian Portuguese features more open, clearly articulated vowels, whereas Continental Portuguese features more closed or reduced vowels, especially in unstressed positions.

Brazilian Portuguese Examples:

1. Ele mora do lado da padaria. (He lives next to the bakery.)
2. Eu estou sempre do seu lado. (I am always on your side.)
3. O carro parou do lado esquerdo. (The car stopped on the left side.)
4. Tente ver o outro lado da situação. (Try to see the other side of the situation.)
5. Deixe o controle do lado da televisão. (Leave the remote next to the television.)

Portuguese (Portugal) Examples:

1. Ele mora ao lado da padaria. (He lives next to the bakery.)
2. Estou sempre do teu lado. (I am always on your side.)
3. O carro parou no lado esquerdo. (The car stopped on the left side.)
4. Tenta ver o outro lado da situação. (Try to see the other side of the situation.)
5. Põe o comando ao lado da televisão. (Put the remote next to the television.)