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

Yes

The word "budista" is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, unstressed vowels (like the "u" or the "i") tend to be more open and clearly articulated. In Continental Portuguese, unstressed vowels are often more "closed" or even neutralized, which can make the word sound more clipped or rapid to a Brazilian ear.

Brazilian Portuguese Examples

1. Ele é um praticante budista muito dedicado. (He is a very dedicated Buddhist practitioner.)
2. Eu gostaria de ler um livro budista sobre meditação. (I would like to read a Buddhist book about meditation.)
3. Ela segue os princípios budistas no dia a dia. (She follows Buddhist principles in her daily life.)
4. A filosofia budista prega a paz. (Buddhist philosophy preaches peace.)
5. Muitas pessoas buscam o caminho budista para encontrar equilíbrio. (Many people seek the Buddhist path to find balance.)

Continental Portuguese Examples

1. Ele é um praticante budista muito dedicado. (He is a very dedicated Buddhist practitioner.)
2. Gostava de ler um livro budista sobre meditação. (I would like to read a Buddhist book about meditation. Note: Portuguese people often use the imperfect indicative "gostava" instead of the conditional "gostaria" for polite requests.)
3. Ela segue os princípios budistas no seu quotidiano. (She follows Buddhist principles in her daily life. Note: "Quotidiano" is more frequently used than "dia a dia" in Portugal.)
4. A filosofia budista prega a paz. (Buddhist philosophy preaches peace.)
5. Muitas pessoas procuram o caminho budista para encontrar o equilíbrio. (Many people look for the Buddhist path to find balance. Note: "Procurar" is the standard verb for "to seek/look for," and the use of the article "o" before "equilíbrio" is very common.)