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

Yes.

The differences are strictly phonetic. In Brazilian Portuguese, unstressed vowels are often "reduced" or neutralized; specifically, the final "o" is typically pronounced as a "u" (sounding like devagarinhu). Additionally, the rhythm of Brazilian Portuguese tends to be more fluid and syllable-timed, whereas Continental Portuguese features more distinct and clearly articulated unstressed vowels.

Brazilian Portuguese Examples

1. Fala devagarinho, por favor. (Speak slowly, please.)
2. O bebê está dormindo, ande devagarinho. (The baby is sleeping, walk slowly.)
3. Vou fazer tudo devagarinho para não errar. (I am going to do everything slowly so as not to make a mistake.)
4. Ele explicou a matéria devagarinho. (He explained the subject slowly.)
5. Pode ir devagarinho, não temos pressa. (You can go slowly, we are in no rush.)

Continental Portuguese Examples

1. Fala devagarinho, por favor. (Speak slowly, please.)
2. O bebé está a dormir, anda devagarinho. (The baby is sleeping, walk slowly.)
3. Vou fazer tudo devagarinho para não errar. (I am going to do everything slowly so as not to make a mistake.)
4. Ele explicou a matéria devagarinho. (He explained the subject slowly.)
5. Podes ir devagarinho, não temos pressa. (You can go slowly, we are in no rush.)