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

No.

While the spelling, meaning, and grammar of "saímos" are identical in both varieties, there are differences in typical usage and pronunciation. In Brazilian Portuguese, while "saímos" is used in writing and formal speech, the conjugation is frequently replaced in colloquial, everyday conversation by the construction "a gente saiu" (using the third-person singular). In Continental Portuguese, "saímos" remains the standard and most common way to express the first-person plural in both formal and informal contexts. Additionally, pronunciation differs: Brazilians tend to pronounce the vowels more openly, whereas Portuguese speakers often reduce vowels and use a more pronounced "sh" sound (/ʃ/) for the "s".

Brazilian Portuguese

1. Saímos para o shopping. (We went to the mall.)
2. Saímos de trem. (We left by train.)
3. Saímos para comer um lanche. (We went out to have a snack.)
4. Saímos para jantar fora. (We went out to eat out.)
5. Saímos de casa bem cedo. (We left the house very early.)

Continental Portuguese

1. Saímos para o centro comercial. (We went to the shopping center.)
2. Saímos de comboio. (We left by train.)
3. Saímos para fazer um lanche. (We went out to have a snack.)
4. Saímos para jantar fora. (We went out to eat out.)
5. Saímos de casa bem cedo. (We left the house very early.)