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

Yes

The word "prender" is identical in meaning, grammar, and spelling in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the pronunciation is generally more "open," with more clearly articulated vowels and a syllable-timed rhythm. In Continental Portuguese, the pronunciation is more "closed," characterized by significant vowel reduction (where unstressed vowels often become nearly silent) and a stress-timed rhythm.

Brazilian Portuguese Examples:

1. A polícia prendeu o bandido ontem à noite. (The police arrested the thief last night.)
2. Você precisa prender o cinto de segurança antes de partir. (You need to fasten your seatbelt before leaving.)
3. O cachorro prendeu o brinquedo na boca. (The dog caught/held the toy in its mouth.)
4. A mala prendeu na porta do ônibus. (The bag got caught in the bus door.)
5. Ela gosta de prender o cabelo com um elástico. (She likes to tie her hair with an elastic.)

Continental Portuguese Examples:

1. A polícia prendeu o criminoso ontem à noite. (The police arrested the criminal last night.)
2. É necessário prender o cinto de segurança antes de partir. (It is necessary to fasten your seatbelt before leaving.)
3. O cão prendeu o osso com os dentes. (The dog caught/held the bone with its teeth.)
4. A mala prendeu na porta do autocarro. (The bag got caught in the bus door.)
5. Ela gosta de prender o cabelo com um elástico. (She likes to tie her hair with an elastic.)