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

Yes

The meaning, spelling, and grammatical conjugation of the word are identical in both dialects. The only difference is in pronunciation: Brazilian Portuguese tends to have more open and clearly articulated vowels, whereas Continental Portuguese often features more closed or "swallowed" unstressed vowels.

Brazilian Portuguese Examples

1. As crianças crescem muito rápido. (Children grow very fast.)
2. Eu cresci em uma cidade pequena. (I grew up in a small town.)
3. A empresa está crescendo muito este ano. (The company is growing a lot this year.)
4. As plantas precisam de sol para crescer. (Plants need sun to grow.)
5. Ele cresceu bastante depois que começou a malhar. (He grew a lot after he started working out.)

Continental Portuguese Examples

1. As crianças crescem muito depressa. (Children grow very fast.)
2. Eu cresci numa cidade pequena. (I grew up in a small town.)
3. A empresa está a crescer muito este ano. (The company is growing a lot this year.)
4. As plantas precisam de sol para crescer. (Plants need sun to grow.)
5. Ele cresceu bastante depois que começou a treinar. (He grew a lot after he started training.)