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

Yes

The spelling, grammar, and core meaning of "banana" are identical in both Brazilian and Continental Portuguese. The primary difference is phonetic. In Brazilian Portuguese, the vowels are generally more open, and the nasalization of the "an" syllable is more distinct and pronounced. In Continental Portuguese, the vowels are more closed, and the pronunciation is more subtle, often sounding more "swallowed" or reduced.

Brazilian Portuguese Examples

1. Eu adoro comer banana no café da manhã. (I love eating banana for breakfast.)
2. Não seja um banana e peça o que você quer. (Don't be a pushover and ask for what you want.)
3. Comprei um cacho de banana na feira hoje. (I bought a bunch of bananas at the market today.)
4. A vitamina de banana é a minha favorita. (Banana smoothie is my favorite.)
5. Ele é um cara muito banana, nunca consegue dizer não. (He is a very weak guy, he can never say no.)

Portuguese Examples

1. Gosto de comer banana com iogurte. (I like to eat banana with yogurt.)
2. Ele é um bocado banana, não tem coragem. (He is a bit of a pushover, he has no courage.)
3. Fui ao supermercado comprar bananas. (I went to the supermarket to buy bananas.)
4. A banana-pão é excelente para fazer fritadas. (Plantain is excellent for making fritters.)
5. Não te comportes como um banana. (Don't behave like a pushover.)