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

No

While the spelling, grammar, and core dictionary meanings of "vivo" are identical in both Brazilian and Continental Portuguese, there is a difference in typical usage and pronunciation.

In Brazilian Portuguese, "vivo" is frequently used as a synonym for "sharp," "quick-witted," or "clever" (e.g., um menino vivo). While an inhabitant of Portugal would understand this, it is not the "typical" way they would express that trait; they would much more naturally use "esperto" or "desperto." Additionally, the pronunciation differs: Brazilians tend to pronounce vowels more openly and clearly, whereas speakers in Portugal often use more closed, reduced, or "clipped" vowels.

Brazilian Portuguese natural usage:

1. Ele ainda está vivo. (He is still alive.)
2. Ele é um menino muito vivo. (He is a very sharp boy.)
3. O azul desse mar é muito vivo. (The blue of this sea is very vivid.)
4. Eu quero me sentir vivo. (I want to feel alive.)
5. O espírito da festa está muito vivo. (The spirit of the party is very much alive.)

Continental Portuguese natural expression:

1. Ele continua vivo. (He remains alive.)
2. Ele é um rapaz muito esperto. (He is a very smart boy.)
3. O azul deste mar é muito vibrante. (The blue of this sea is very vibrant.)
4. Quero sentir-me vivo. (I want to feel alive.)
5. O espírito da festa está muito presente. (The spirit of the party is very much present.)