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

No.

While the fundamental meaning of "correndo" remains the same—referring to the act of running, being in a hurry, or something being in progress—the typical usage differs significantly between the two dialects.

In Brazilian Portuguese, the gerund ("correndo") is the standard way to express an ongoing action (the continuous aspect). For example, "Estou correndo" (I am running).

In Continental Portuguese, the gerund is rarely used to express ongoing actions. Instead, the standard construction is the preposition a + infinitive. A person from Portugal would typically say "Estou a correr" (I am running). Furthermore, while "correndo" can be used in Portugal to mean "in progress" (e.g., "o tempo está correndo"), the grammatical structure surrounding it usually shifts to the "a + infinitive" pattern.

Brazilian Portuguese Examples

1. Eu estou correndo no parque agora. (I am running in the park right now.)
2. Ele está correndo para não perder o ônibus. (He is running so he doesn't miss the bus.)
3. A gente está correndo muito hoje. (We are running/in a hurry a lot today.)
4. O risco está correndo muito alto. (The risk is running/becoming very high.)
5. Ela passou a tarde correndo atrás do cachorro. (She spent the afternoon running after the dog.)

Continental Portuguese Examples

1. Eu estou a correr no parque agora. (I am running in the park right now.)
2. Ele está a correr para não perder o autocarro. (He is running so he doesn't miss the bus.)
3. Nós estamos a correr muito hoje. (We are running/in a hurry a lot today.)
4. O risco está a correr demasiado alto. (The risk is running/becoming too high.)
5. Ela passou a tarde a correr atrás do cão. (She spent the afternoon running after the dog.)