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

No.

While the spelling of "Enzo" is identical in both varieties, there is a difference in typical usage. In Brazil, "Enzo" has become a widespread cultural meme and slang term used to stereotype a specific type of modern, often pampered, child (part of the "Enzo and Valentina" phenomenon). In Portugal, the word is used strictly as a proper name without this sociological connotation. Additionally, there is a slight difference in pronunciation; Brazilian Portuguese often features more nasalization and a different vowel openness in the "en" syllable compared to the more closed pronunciation in Portugal.

Brazilian Portuguese

1. O Enzo está dormindo. (Enzo is sleeping.)
2. O Enzo é um nome bem comum hoje em dia. (Enzo is a very common name nowadays.)
3. Eu conheci o Enzo ontem. (I met Enzo yesterday.)
4. Aquele garoto é muito Enzo. (That kid is such an "Enzo" [referring to the stereotype].)
5. O Enzo Gabriel é o nome dele. (Enzo Gabriel is his name.)

Portuguese (Portugal)

1. O Enzo está a dormir. (Enzo is sleeping.)
2. O Enzo é um nome muito comum atualmente. (Enzo is a very common name currently.)
3. Eu conheci o Enzo ontem. (I met Enzo yesterday.)
4. Aquele miúdo chama-se Enzo. (That kid is named Enzo.)
5. O nome dele é Enzo Gabriel. (His name is Enzo Gabriel.)