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

No

While the fundamental meaning of "pegar" (to grab, to catch, to take) is shared, the primary difference lies in usage and lexical preference. In Brazilian Portuguese, "pegar" is a highly versatile "catch-all" verb used for almost everything involving catching or taking something (transportation, objects, illnesses, or even people). In Continental Portuguese, the verb "apanhar" is much more natural and common for the acts of catching a bus, catching a cold, or catching a moving object. While a Portuguese person can use "pegar" for the physical act of grasping an object, "apanhar" is the standard for "catching"-related contexts. Additionally, the nouns used alongside these verbs often differ (e.g., ônibus vs. autocarro).

Brazilian Portuguese Usage

1. Eu vou pegar o ônibus agora. (I am going to catch the bus now.)
2. Ele pegou um resfriado. (He caught a cold.)
3. Você pode pegar o meu celular para mim? (Can you grab my cell phone for me?)
4. Eu peguei ela na festa. (I hooked up with her at the party.)
5. Vou pegar um táxi. (I'm going to take a taxi.)

Continental Portuguese Usage

1. Eu vou apanhar o autocarro agora. (I am going to catch the bus now.)
2. Ele apanhou uma constipação. (He caught a cold.)
3. Podes apanhar o meu telemóvel para mim? (Can you grab my mobile phone for me?)
4. Ele apanhou o comboio. (He caught the train.)
5. Vou apanhar um táxi. (I am going to take a taxi.)