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

Yes

The word "vamos" is grammatically, semantically, and orthographically identical in both Brazilian and Continental Portuguese. It functions as the first-person plural of the verb ir (to go) and as an imperative/hortatory particle (meaning "let's"). The differences are strictly phonetic:

• Pronunciation: In Brazilian Portuguese, the "s" is typically pronounced as [s] (though [ʃ] is common in Rio de Janeiro and other coastal regions), and the final "os" is often reduced to a neutral "u" sound ([ˈvamu]). In Continental Portuguese, the "s" is almost always pronounced as a "sh" sound ([ʃ]), and the vowels are generally more closed and less nasalized.

Brazilian Portuguese

1. Vamos ao shopping hoje? (Shall we go to the mall today?)
2. Vamos lá, você consegue! (Come on, you can do it!)
3. Vamos pegar um ônibus. (Let's take a bus.)
4. Vamos usar o celular. (Let's use the cell phone.)
5. Vamos comer uma pizza? (Let's eat a pizza?)

Continental Portuguese

1. Vamos ao centro comercial hoje? (Shall we go to the shopping center today?)
2. Vamos lá, tu consegues! (Come on, you can do it!)
3. Vamos apanhar um autocarro. (Let's take a bus.)
4. Vamos usar o telemóvel. (Let's use the mobile phone.)
5. Vamos comer uma pizza? (Let's eat a pizza?)