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

Yes.

The word "sentir" is identical in meaning, spelling, and conjugation in both Brazilian and Continental Portuguese. The differences are strictly phonological:

• Vowels: Brazilian Portuguese tends to have more open, "syllabic" vowels where each vowel is clearly pronounced. Continental Portuguese features "vowel reduction," where unstressed vowels are often shortened or nearly silent.
• The "r" sound: In many Brazilian dialects, the final "r" is a soft, aspirated sound (similar to the "h" in "house"). In Portugal, the "r" is typically more guttural and uvular.

Brazilian Portuguese Examples

• Eu me sinto muito cansado hoje. (I feel very tired today.)
• Você está sentindo calor? (Are you feeling hot?)
• A gente sente muita saudade de você. (We miss you a lot.)
• Eu sinto muito pelo que aconteceu. (I am very sorry for what happened.)
• Eu senti um cheiro de queimado. (I smelled something burning.)

Continental Portuguese Examples

• Sinto-me muito cansado hoje. (I feel very tired today.)
• Estás a sentir calor? (Are you feeling hot?)
• Nós sentimos muita saudade de ti. (We miss you a lot.)
• Sinto muito pelo que aconteceu. (I am very sorry for what happened.)
• Senti um cheiro de queimado. (I smelled something burning.)