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

Yes

The word "mesma" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. It functions as an adjective (meaning "same") or an intensifier (meaning "myself," "even," or "really"). The only differences are found in pronunciation: Brazilian Portuguese tends to have more open vowels and a more melodic rhythm, whereas Continental Portuguese features more vowel reduction (where unstressed vowels are shortened or nearly silent) and a more "closed" phonetic delivery.

Brazilian Portuguese Examples

1. Eu mesma preparei o jantar. (I prepared dinner myself.)
2. Ela sempre diz a mesma coisa. (She always says the same thing.)
3. Você viu a mesma foto no celular? (Did you see the same photo on the cell phone?)
4. É a mesma rua de sempre. (It is the same street as always.)
5. O ônibus passou na mesma hora. (The bus passed at the same time.)

Continental Portuguese Examples

1. Eu mesma preparei o jantar. (I prepared dinner myself.)
2. Ela sempre diz a mesma coisa. (She always says the same thing.)
3. Viste a mesma foto no telemóvel? (Did you see the same photo on the cell phone?)
4. É a mesma rua de sempre. (It is the same street as always.)
5. O autocarro passou na mesma hora. (The bus passed at the same time.)