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

No

The difference lies in the spelling and pronunciation when referring to the subway system. In Brazilian Portuguese, the word for the subway is spelled metrô (with a circumflex accent), which indicates a closed "o" sound. In Continental Portuguese, the word is spelled metro (without an accent). When referring to the unit of measurement (the meter), the spelling and meaning are identical in both varieties.

Brazilian Portuguese examples:

1. Eu peguei o metrô para chegar mais rápido. (I took the subway to arrive faster.)
2. O metrô de São Paulo estava muito lotado. (The São Paulo subway was very crowded.)
3. Onde fica a estação de metrô mais próxima? (Where is the nearest subway station?)
4. O metrô parou por causa de um problema técnico. (The subway stopped because of a technical problem.)
5. Eu comprei um tecido de um metro de comprimento. (I bought a fabric one meter in length.)

Continental Portuguese examples:

1. Eu apanhei o metro para chegar mais depressa. (I caught the subway to arrive faster.)
2. O metro de Lisboa estava muito cheio. (The Lisbon subway was very full.)
3. Onde fica a estação de metro mais próxima? (Where is the nearest subway station?)
4. O metro parou devido a um problema técnico. (The subway stopped due to a technical problem.)
5. Eu comprei um tecido de um metro de comprimento. (I bought a fabric one meter in length.)