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

No

The spelling and core meaning of "veste" (the third-person singular present of the verb vestir) are the same in both varieties. However, the answer is No because there are significant differences in grammar and typical usage:

• Grammar (Clitic Placement): When used reflexively (to dress oneself), Brazilian Portuguese (BP) typically uses proclisis, placing the pronoun before the verb (ele se veste). In Continental Portuguese (CP), enclisis is the standard, placing the pronoun after the verb (ele veste-se).
• Typical Usage (Lexical Preference): While "veste" is grammatically correct in Brazil, it is much more natural for a Brazilian to use the verb usar (to use/to wear) when describing what someone is currently wearing. In Portugal, vestir remains the standard verb for this context.

Brazilian Portuguese Examples

1. Ele se veste muito rápido. (He dresses very quickly.)
2. Ela se veste com muita elegância. (She dresses with great elegance.)
3. O menino se veste sozinho agora. (The boy dresses himself alone now.)
4. Ela veste sempre roupas pretas. (She always wears black clothes.)
5. Ele está usando uma camisa azul. (He is wearing a blue shirt.)

Continental Portuguese Examples

1. Ele veste-se muito rápido. (He dresses very quickly.)
2. Ela veste-se com muita elegância. (She dresses with great elegance.)
3. O menino veste-se sozinho agora. (The boy dresses himself alone now.)
4. Ela veste sempre roupas pretas. (She always wears black clothes.)
5. Ele está a vestir uma camisa azul. (He is wearing a blue shirt.)