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

No

While the spelling and primary definition of "bombeiro" are the same in both dialects, there is a difference in typical usage. In Brazil, "bombeiro" is frequently used as a colloquial shorthand for bombeiro hidráulico (plumber). In Portugal, "bombeiro" refers almost exclusively to a member of the fire service; using the word to refer to a plumber would be confusing, as a Portuguese person would use the word canalizador.

Brazilian Portuguese

1. O bombeiro apagou o incêndio rapidamente. (The firefighter put out the fire quickly.)
2. Preciso chamar um bombeiro para consertar este vazamento. (I need to call a plumber to fix this leak.)
3. O bombeiro salvou o cachorro que estava na árvore. (The firefighter saved the dog that was in the tree.)
4. Ele trabalha como bombeiro há muitos anos. (He has worked as a firefighter for many years.)
5. Os bombeiros estão no treinamento hoje. (The firefighters are in training today.)

European Portuguese

1. O bombeiro apagou o incêndio rapidamente. (The firefighter put out the fire quickly.)
2. Preciso chamar um canalizador para consertar este vazamento. (I need to call a plumber to fix this leak.)
3. O bombeiro salvou o cão que estava na árvore. (The firefighter saved the dog that was in the tree.)
4. Ele trabalha como bombeiro há muitos anos. (He has worked as a firefighter for many years.)
5. Os bombeiros estão no treino hoje. (The firefighters are in training today.)