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

No

While the primary meanings of "vaso" (a plant pot, a decorative vase, or a blood vessel) are the same in both dialects, there is a difference in typical usage regarding plumbing. In Brazil, "vaso" is a very common shorthand for vaso sanitário (the toilet). In Portugal, the standard word for the toilet is "sanita," and using "vaso" in that context would almost always be interpreted as a container for plants or a decorative vessel.

Brazilian Portuguese

1. O vaso de plantas está quebrado. (The flowerpot is broken.)
2. O vaso está entupido. (The toilet is clogged.)
3. Ela comprou um vaso de cristal lindo. (She bought a beautiful crystal vase.)
4. Ele tem problemas nos vasos sanguíneos. (He has problems with his blood vessels.)
5. Coloque as flores naquele vaso azul. (Put the flowers in that blue vase.)

European Portuguese

1. O vaso da planta está partido. (The plant pot is broken.)
2. A sanita está entupida. (The toilet is clogged.)
3. Ela comprou um vaso de cristal muito bonito. (She bought a very beautiful crystal vase.)
4. Ele tem problemas nos vasos sanguíneos. (He has problems with his blood vessels.)
5. Coloca as flores naquele vaso azul. (Put the flowers in that blue vase.)