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

No

While the core meaning of "gordo" (referring to biological fat or weight) remains the same, there is a difference in typical usage and pronunciation:

• Usage: In Brazilian Portuguese, "gordo" is frequently used colloquially to describe the thickness or diameter of inanimate objects (e.g., a "gordo" pencil or a "gordo" line). In Continental Portuguese, "grosso" is the standard and much more natural word for thickness, whereas "gordo" is almost exclusively reserved for living beings or fat content.
• Pronunciation: In Brazil, the unstressed final "o" is pronounced as a closed [u] (sounding like "gordu"). In Portugal, the final "o" is reduced to a much shorter, neutral [ɐ] (sounding like "gordɐ").

Brazilian Portuguese

1. O meu cachorro é muito gordo. (My dog is very fat.)
2. Ele comprou uma caneta bem gorda. (He bought a very thick pen.)
3. Aquele menino é muito gordinho. (That boy is very chubby.)
4. Eu não gosto de carne muito gorda. (I don't like very fatty meat.)
5. O dedo dele é muito gordo. (His finger is very fat.)

Portuguese (Portugal)

1. O meu cão é muito gordo. (My dog is very fat.)
2. Ele comprou uma caneta bem grossa. (He bought a very thick pen.)
3. Aquele miúdo é muito gordinho. (That kid is very chubby.)
4. Eu não gosto de carne muito gorda. (I don't like very fatty meat.)
5. O dedo dele é muito grosso. (His finger is very thick.)