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

Yes

The word "peso" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the vowels tend to be more open and the rhythm is more syllable-timed, making the word sound more melodic. In Continental Portuguese, the unstressed vowels are more closed or even reduced (almost disappearing), and the rhythm is more stress-timed, which gives the word a more clipped, staccato sound.

Brazilian Portuguese Usage

1. Eu preciso perder peso. (I need to lose weight.)
2. O peso dessa mala está muito grande. (The weight of this suitcase is very large.)
3. Isso me traz um peso enorme na consciência. (This brings a huge weight to my conscience.)
4. O peso da mochila me incomoda. (The weight of the backpack bothers me.)
5. Ele está tentando controlar o peso. (He is trying to control his weight.)

Continental Portuguese Usage

1. Eu preciso de perder peso. (I need to lose weight.)
2. O peso desta mala é excessivo. (The weight of this suitcase is excessive.)
3. Isso pesa-me muito na consciência. (This weighs heavily on my conscience.)
4. O peso desta mochila incomoda-me. (The weight of this backpack bothers me.)
5. Ele está a controlar o peso. (He is controlling his weight.)