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

No

While the fundamental definition of "realizar" (to achieve, to fulfill, or to perform) is the same in both dialects, there is a difference in typical usage and register. In Brazilian Portuguese, "realizar" is frequently used in professional, academic, or administrative contexts to mean "to carry out," "to execute," or "to conduct" (e.g., conducting a study or a task). While this is grammatically correct in Portugal, a speaker from Portugal would more naturally use the verb "fazer" (to do) for these routine actions, or the more formal "efetuar" (to effect/perform) for medical or official procedures. In Portugal, "realizar" is more often reserved for the sense of "achieving" or "fulfilling" something profound, such as a dream or a long-term goal.

Brazilian Portuguese

1. Eu realizei meu sonho de viajar pelo mundo. (I fulfilled my dream of traveling the world.)
2. A empresa realizou uma pesquisa de mercado. (The company conducted market research.)
3. Ele realizou a tarefa muito rapidamente. (He performed the task very quickly.)
4. Precisamos realizar esse projeto até amanhã. (We need to carry out this project by tomorrow.)
5. O médico realizou o procedimento com sucesso. (The doctor performed the procedure successfully.)

Continental Portuguese

1. Eu realizei o meu sonho de viajar pelo mundo. (I fulfilled my dream of traveling the world.)
2. A empresa fez uma sondagem de mercado. (The company conducted a market poll.)
3. Ele fez a tarefa muito rapidamente. (He did the task very quickly.)
4. Temos de fazer este projeto até amanhã. (We have to do this project by tomorrow.)
5. O médico efetuou o procedimento com sucesso. (The doctor performed the procedure successfully.)