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

Yes

The word "organizo" is identical in meaning, spelling, and grammatical conjugation in both Brazilian and Continental Portuguese. The only difference lies in pronunciation (phonology). In Brazilian Portuguese, the vowels tend to be more open and the rhythm is more musical/syllable-timed. In Continental Portuguese, vowels are often more closed or reduced (sometimes almost swallowed), and the rhythm is more stress-timed, which can make the unstressed syllables sound much shorter.

Brazilian Portuguese Examples

1. Eu organizo meu quarto todo sábado. (I organize my room every Saturday.)
2. Eu organizo os arquivos no computador. (I organize the files on the computer.)
3. Eu organizo minha agenda de trabalho. (I organize my work schedule.)
4. Eu organizo as malas para a viagem. (I organize the suitcases for the trip.)
5. Eu organizo os documentos da empresa. (I organize the company documents.)

Continental Portuguese Examples

1. Eu organizo o meu quarto todos os sábados. (I organize my room every Saturday.)
2. Eu organizo os ficheiros no computador. (I organize the files on the computer.)
3. Eu organizo a minha agenda de trabalho. (I organize my work schedule.)
4. Eu organizo as malas para a viagem. (I organize the suitcases for the trip.)
5. Eu organizo os documentos da empresa. (I organize the company documents.)