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

Yes

The word "Laura" is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. The only difference is the pronunciation. In Brazilian Portuguese, the diphthong "au" is typically pronounced with more open, resonant vowels (closer to "LÁU-ra"). In Continental Portuguese, the vowels are often more closed or reduced, resulting in a slightly more subtle or "clipped" sound.

Brazilian Portuguese Examples

1. A Laura pegou o ônibus. (Laura took the bus.)
2. A Laura está estudando muito. (Laura is studying a lot.)
3. Você viu a Laura no shopping? (Did you see Laura at the mall?)
4. O celular da Laura quebrou. (Laura's cell phone broke.)
5. A Laura gosta de sorvete. (Laura likes ice cream.)

Continental Portuguese Examples

1. A Laura apanhou o autocarro. (Laura caught the bus.)
2. A Laura está a estudar muito. (Laura is studying a lot.)
3. Já viste a Laura no centro comercial? (Have you seen Laura at the mall?)
4. O telemóvel da Laura avariou. (Laura's cell phone broke/malfunctioned.)
5. A Laura gosta de gelado. (Laura likes ice cream.)