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

Yes

The word "fome" is identical in spelling, meaning, and grammatical usage in both Brazilian and European Portuguese. The only difference is the pronunciation of the final unstressed "e." In Brazilian Portuguese, the "e" is typically pronounced as a clear [i] (sounding like "fo-mee"). In European Portuguese, the "e" is more closed and neutralized, often sounding like a short [ɨ] or nearly disappearing (sounding more like "fom(e)").

Brazilian Portuguese Examples

1. Estou com muita fome. (I am very hungry.)
2. Você está com fome? (Are you hungry?)
3. Bateu uma fome do nada! (I suddenly got hungry!)
4. A fome é um problema sério. (Hunger is a serious problem.)
5. Ele está com uma fome de leão. (He is hungry as a lion.)

European Portuguese Examples

1. Estou com muita fome. (I am very hungry.)
2. Estás com fome? (Are you hungry?)
3. Sinto uma fome enorme. (I feel an enormous hunger.)
4. A fome é um problema grave. (Hunger is a serious problem.)
5. Morri de fome. (I am starving / I died of hunger.)