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

No.

While the spelling and the fundamental meaning of the word chego (the first-person singular present indicative of the verb chegar) are the same, there is a difference in typical usage and grammar (specifically regência verbal, or verb complementation).

In Brazil, it is standard and natural to use the preposition em (and its contractions no/na) after the verb. In Portugal, the standard and natural usage is with the preposition a (and its contractions ao/à). Additionally, certain nouns used in the context of "arriving" (like "bus stop" or "mall") differ between the two countries.

Brazilian Portuguese Examples

1. Eu chego em casa às oito horas. (I arrive home at eight o'clock.)
2. Eu chego no trabalho muito cedo. (I arrive at work very early.)
3. Eu chego no shopping agora. (I arrive at the mall now.)
4. Eu chego no Rio de Janeiro amanhã. (I arrive in Rio de Janeiro tomorrow.)
5. Eu chego no ponto de ônibus em breve. (I arrive at the bus stop shortly.)

Continental Portuguese Examples

1. Eu chego a casa às oito horas. (I arrive home at eight o'clock.)
2. Eu chego ao trabalho muito cedo. (I arrive at work very early.)
3. Eu chego ao centro comercial agora. (I arrive at the mall now.)
4. Eu chego ao Rio de Janeiro amanhã. (I arrive in Rio de Janeiro tomorrow.)
5. Eu chego à paragem de autocarro em breve. (I arrive at the bus stop shortly.)