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

Yes

The word "bastante" is semantically, grammatically, and orthographically identical in both Brazilian and Continental Portuguese. In both dialects, it functions as an adverb (meaning "very," "quite," or "sufficiently") and as an adjective or determiner (meaning "sufficient," "much," or "many"). The primary differences lie in pronunciation—specifically the reduction of unstressed vowels in Portugal—and the stylistic preference for using synonyms like "muito" in certain contexts in Portugal.

Brazilian Portuguese Examples

1. Este filme é bastante interessante. (This movie is quite interesting.)
2. Eu tenho bastante trabalho hoje. (I have a lot of work today.)
3. Ela fala bastante rápido. (She speaks quite fast.)
4. Comemos bastante no jantar. (We ate plenty at dinner.)
5. Eu tenho bastante comida na geladeira. (I have plenty of food in the fridge.)

Portuguese (Portugal) Examples

1. Este filme é muito interessante. (This movie is very interesting.)
2. Tenho muito trabalho hoje. (I have a lot of work today.)
3. Ela fala muito depressa. (She speaks very quickly.)
4. Comemos muito ao jantar. (We ate a lot at dinner.)
5. Tenho muita comida no frigorífico. (I have plenty of food in the fridge.)