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

No

While the word "bolsa" shares the same core meanings in both dialects—such as "stock market" (Bolsa de Valores) or "scholarship" (bolsa de estudos)—there is a significant difference in typical usage regarding physical objects. In Brazil, "bolsa" is the standard, everyday word for a woman's handbag or purse. In Portugal, while "bolsa" can be used to refer to a small pouch or clutch, the word "mala" is much more commonly and naturally used to refer to a handbag or a larger bag. Additionally, when discussing the material of the bag, Brazilians typically use couro (leather), whereas Portuguese people more naturally use pele (skin/leather).

Brazilian Portuguese

1. Esqueci minha bolsa no ônibus. (I forgot my purse on the bag.)
2. Ela comprou uma bolsa de couro nova. (She bought a new leather handbag.)
3. Essa bolsa combina com o seu vestido. (That handbag matches your dress.)
4. Preciso de uma bolsa maior para viajar. (I need a larger bag to travel.)
5. Onde você comprou essa bolsa? (Where did you buy that handbag?)

Continental Portuguese

1. Esqueci a minha mala no autocarro. (I forgot my handbag on the bus.)
2. Ela comprou uma mala de pele nova. (She bought a new leather handbag.)
3. Essa mala combina com o teu vestido. (That handbag matches your dress.)
4. Preciso de uma mala maior para viajar. (I need a larger bag to travel.)
5. Onde é que compraste essa mala? (Where did you buy that handbag?)