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

No.

While the spelling and the fundamental meaning of "vive" (the third-person singular present indicative of the verb viver) remain the same, there are differences in usage and pronunciation.

• Usage/Grammar: In Brazilian Portuguese, when expressing a habitual or continuous action, it is natural to use the gerund (e.g., vive correndo). In Continental Portuguese, the standard construction uses the preposition a followed by the infinitive (e.g., vive a correr).
• Pronunciation: Brazilian Portuguese tends to pronounce the vowels more openly and clearly. In Continental Portuguese, unstressed vowels are often reduced or "swallowed," making the second syllable sound much shorter and more closed.

Brazilian Portuguese Examples

1. Ele vive reclamando de tudo. (He is always complaining about everything.)
2. O cachorro vive correndo no jardim. (The dog is always running in the garden.)
3. Ela vive viajando pela Europa. (She is always traveling through Europe.)
4. Ele vive perdendo o celular. (He is always losing his cell phone.)
5. Ela vive falando sozinha. (She is always talking to herself.)

Continental Portuguese Examples

1. Ele vive a reclamar de tudo. (He is always complaining about everything.)
2. O cão vive a correr no jardim. (The dog is always running in the garden.)
3. Ela vive a viajar pela Europa. (She is always traveling through Europe.)
4. Ele vive a perder o telemóvel. (He is always losing his mobile phone.)
5. Ela vive a falar sozinha. (She is always talking to herself.)