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

Yes

The spelling, meaning, and grammatical gender of "pêssego" are identical in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels are typically more open and the syllables are pronounced with more equal duration (syllable-timed). In Continental Portuguese, unstressed vowels are much more reduced or "clipped," often sounding almost like a "shwa" [ə] or disappearing entirely, resulting in a more stress-timed rhythm.

Brazilian Portuguese Examples

1. Eu amo comer pêssego no café da manhã. (I love eating peach for breakfast.)
2. Esse suco de pêssego está muito gostoso. (This peach juice is very tasty.)
3. Comprei um pêssego bem docinho. (I bought a very sweet peach.) . O pêssego está na época. (The peach is in season.)
4. Eu fiz uma torta de pêssego. (I made a peach pie.)

Continental Portuguese Examples

1. Eu adoro comer pêssego no pequeno-almoço. (I love eating peach for breakfast.)
2. Este sumo de pêssego está muito bom. (This peach juice is very good.)
3. Comprei um pêssego muito doce. (I bought a very sweet peach.)
4. O pêssego está na época. (The peach is in season.)
5. Eu fiz uma tarte de pêssego. (I made a peach tart.)