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

No.

While the spelling, grammar, and core meaning of "busca" are identical in both dialects, there are differences in typical usage and pronunciation.

In terms of usage, while a Portuguese person understands "busca," they more frequently use synonyms like procura (for searching) or apanhar (for picking up/fetching) in everyday conversation. In Brazil, "busca" and the verb "buscar" are the standard, go-to terms for these actions.

In terms of pronunciation, Brazilian Portuguese features an open, clear vowel sound at the end of the word (/ˈbus.kɐ/), whereas Continental Portuguese heavily reduces unstressed vowels, making the final "a" sound like a nearly silent, muffled schwa (/ˈbus.kə/).

Brazilian Portuguese (Typical Usage)

1. A busca por um novo emprego é cansativa. (The search for a new job is tiring.)
2. Eu vou buscar você na estação de trem. (I am going to pick you up at the train station.)
3. A busca de informações na internet é muito rápida. (Searching for information on the internet is very fast.)
4. Ele está em busca de uma solução para o problema. (He is in search of a solution to the problem.)
5. A busca do tesouro começou ontem. (The search for the treasure began yesterday.)

Continental Portuguese (Natural Alternatives)

1. A procura por um novo emprego é cansativa. (The search for a new job is tiring.)
2. Eu vou apanhar-te na estação de comboio. (I am going to pick you up at the train station.)
3. A pesquisa de informações na internet é muito rápida. (Searching for information on the internet is very fast.)
4. Ele está à procura de uma solução para o problema. (He is in search of a solution to the problem.)
5. A procura do tesouro começou ontem. (The search for the treasure began yesterday.)