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

No

The spelling, meaning, and grammatical functions of "logo" are identical in both Brazilian and Continental Portuguese. However, there are differences in typical usage. In Brazil, "logo" is frequently used in the temporal expression "logo mais" to mean "later" or "soon," a construction that is much less common in Portugal. Additionally, while "logo" is used in Portugal to refer to a graphic emblem, "logótipo" is the much more frequent and standard term used by Portuguese speakers.

Brazilian Portuguese Examples

1. Eu logo volto. (I will be back soon.)
2. Logo, terminaremos o projeto. (Therefore, we will finish the project.)
3. A gente se vê logo. (We will see each other soon.)
4. Logo mais eu te ligo. (I will call you later/soon.)
5. O logo da empresa é muito bonito. (The company's logo is very beautiful.)

Continental Portuguese Examples

1. Volto logo. (I will be back soon.)
2. Portanto, terminaremos o projeto. (Therefore, we will finish the project.)
3. Vemo-nos logo. (We will see each other soon.)
4. Ligo-te daqui a pouco. (I will call you in a little while.)
5. O logótipo da empresa é muito bonito. (The company's logo is very beautiful.)