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

Yes

The word "correr" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The differences are strictly phonetic. In Brazilian Portuguese, the pronunciation of the final "r" is highly regional, often being a retroflex "r" (common in the interior of the country) or a glottal "h" sound (common in Rio de Janeiro). In Continental Portuguese, the "r" is typically an alveolar tap, and the vowels are often more "closed" or reduced, resulting in a more clipped pronunciation.

Brazilian Portuguese

1. Eu gosto de correr no parque. (I like to run in the park.)
2. Ele está correndo para não perder o ônibus. (He is running so as not to miss the bus.)
3. Você corre muito rápido! (You run very fast!)
4. A gente precisa correr para chegar a tempo. (We need to run to arrive on time.)
5. O tempo está correndo. (Time is running.)

Continental Portuguese

1. Eu gosto de correr no parque. (I like to run in the park.)
2. Ele está a correr para não perder o autocarro. (He is running so as not to miss the bus.)
3. Tu corres muito depressa! (You run very fast!)
4. Temos de correr para chegar a tempo. (We have to run to arrive on time.)
5. O tempo está a correr. (Time is running.)