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

No

The difference between the two variants lies in the spelling and pronunciation. In Brazilian Portuguese, the word is spelled with a circumflex accent (karaokê) to indicate an open "e" sound at the end of the word. In Continental Portuguese, the accent is omitted (karaoke), following the international spelling. Regarding pronunciation, Brazilians pronounce the final syllable with a strong, open "e" ([ɛ]), whereas Portuguese speakers typically use a closed or reduced "e" ([e] or [ɨ]).

Brazilian Portuguese Examples

1. Vamos fazer um karaokê hoje à noite? (Let's do karaoke tonight?)
2. Eu adoro cantar no karaokê com meus amigos. (I love singing karaoke with my friends.)
3. O karaokê estava muito lotado. (The karaoke bar was very crowded.)
4. Ele é muito bom no karaokê. (He is very good at karaoke.)
5. Meus amigos e eu fomos ao karaokê no fim de semana. (My friends and I went to karaoke over the weekend.)

Portuguese (PT-PT) Examples

1. Vamos ao karaoke hoje à noite? (Shall we go to karaoke tonight?)
2. Eu adoro cantar no karaoke com os meus amigos. (I love singing karaoke with my friends.)
3. O karaoke estava muito cheio. (The karaoke bar was very full.)
4. Ele é muito bom no karaoke. (He is very good at karaoke.)
5. Eu e os meus amigos fomos ao karaoke no fim de semana. (My friends and I went to karaoke over the weekend.)