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

No

While the meaning and spelling of the word "foge" are identical in both dialects, there is a difference in typical usage regarding its grammatical mood. In Brazilian Portuguese, "foge" is most naturally used as the third-person singular indicative (stating that someone/something is fleeing). When a Brazilian wants to issue a command (the imperative mood) to "você," they typically use the form "fuja." In Continental Portuguese, "foge" is the standard and natural way to issue a command to "tu" (the common second-person singular).

Brazilian Portuguese (Indicative usage)

1. O gato foge quando vê o cachorro. (The cat runs away when it sees the dog.)
2. Ele foge das responsabilidades. (He runs away from responsibilities.)
3. A criança foge do médico. (The child runs away from the doctor.)
4. O ladrão foge pela janela. (The thief escapes through the window.)
5. Ela foge do assunto. (She runs away from the subject.)

Portuguese (Imperative usage)

1. Foge do cão! (Run from the dog!)
2. Foge das responsabilidades! (Run from responsibilities!)
3. Foge do médico! (Run from the doctor!)
4. Foge pela janela! (Escape through the window!)
5. Foge do assunto! (Run from the subject!)