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

Yes

The spelling, grammar, and fundamental meaning of the word "centro" are identical in both Brazilian and Continental Portuguese. The only differences lie in pronunciation and phonetic rhythm.

In Brazilian Portuguese, the pronunciation is generally more "open" and melodic. The vowels are articulated more clearly, and the rhythm is "syllable-timed," meaning each syllable receives a relatively similar amount of weight. In Continental Portuguese, the pronunciation is more "closed" and "clipped." The rhythm is "stress-timed," which means unstressed syllables—especially the final "o" in "centro"—are much more reduced, often sounding like a very short, nearly silent, or extremely closed "u" sound.

Brazilian Portuguese Examples

1. Eu vou ao centro agora à tarde. (I am going downtown this afternoon.)
2. O centro da cidade é muito movimentado. (The city center is very busy.) . Eles inauguraram um novo centro de treinamento. (They inaugurated a new training center.)
3. O shopping fica bem no centro. (The mall is located right downtown.)
4. O ponto central fica no centro do círculo. (The central point is in the center of the circle.)

Continental Portuguese Examples

1. Vou ao centro tratar de uns papéis. (I am going downtown to handle some papers.)
2. O centro da cidade tem muita história. (The city center has a lot of history.)
3. O centro comercial é enorme. (The shopping center is huge.)
4. O centro de saúde está fechado. (The health center is closed.)
5. Coloque o objeto no centro da mesa. (Place the object in the center of the table.)