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

No.

While the spelling, meaning, and grammatical definition of "peço" are identical in both dialects, there are differences in typical usage, specifically regarding syntax (the placement of pronouns) and the nouns that commonly accompany the verb. In Brazil, it is common to place pronouns before the verb (proclisis) and use the plural "desculpas," whereas in Portugal, pronouns are traditionally placed after the verb (enclisis) and the singular "desculpa" is more standard.

Brazilian Portuguese Examples

1. Eu te peço um favor. (I ask you a favor.)
2. Peço desculpas pelo atraso. (I apologize for the delay.)
3. Peço que você me avise. (I ask that you let me know.)
4. Peço para você esperar um pouco. (I ask you to wait a little bit.)
5. Peço um café, por favor. (I'll have a coffee, please.)

Continental Portuguese Examples

1. Peço-te um favor. (I ask you a favor.)
2. Peço desculpa pelo atraso. (I apologize for the delay.)
3. Peço que me avises. (I ask that you let me know.)
4. Peço que esperes um pouco. (I ask that you wait a little bit.)
5. Peço um café, por favor. (I'll have a coffee, please.)