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

No

While the spelling and conjugation of "pegou" are the same in both dialects, there are significant differences in usage and vocabulary preference.

In Brazilian Portuguese, "pegar" is a highly versatile "utility verb" used for a wide range of actions, including catching transport, catching an illness, grabbing an object, or even understanding a concept. In Continental Portuguese, while "pegar" is used for physical grabbing, the verb apanhar is much more frequently used for catching things in motion (like a ball), catching transport (like a bus), or catching an illness. Furthermore, where a Brazilian might use "pegar" to mean "to understand," a Portuguese person would more naturally use perceber.

Brazilian Portuguese Usage

1. Ele pegou o ônibus agora. (He caught the bus just now.)
2. Ela pegou o telefone que caiu. (She grabbed the phone that fell.)
3. Você pegou o que eu disse? (Did you get what I said?)
4. Ele pegou uma gripe forte. (He caught a bad flu.)
5. O gato pegou o rato. (The cat caught the mouse.)

Continental Portuguese Usage

1. Ele apanhou o autocarro agora. (He caught the bus just now.)
2. Ela apanhou o telefone que caiu. (She grabbed/caught the phone that fell.)
3. Tu percebeste o que eu disse? (Did you understand what I said?)
4. Ele apanhou uma constipação forte. (He caught a bad cold.)
5. O gato apanhou o rato. (The cat caught the mouse.)