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

Yes

The spelling, meaning, and grammatical classification of the word are identical in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the vowels tend to be more open, and the rhythm is more syllable-timed. In Continental Portuguese, unstressed vowels are often more closed or even reduced to nearly inaudible sounds, and the prosody (the rhythm and intonation of the sentence) differs.

Brazilian Portuguese Examples

1. Eu quero muito viajar para o México. (I really want to travel to Mexico.)
2. A comida do México é muito gostosa. (Mexican food is very tasty.)
3. A gente vai visitar o México nas férias. (We are going to visit Mexico on vacation.)
4. Você já foi ao México? (Have you ever been to Mexico?)
5. O México é um lugar muito legal. (Mexico is a very cool place.)

Continental Portuguese Examples

1. Eu pretendo viajar ao México. (I intend to travel to Mexico.)
2. A comida do México é bastante saborosa. (Mexican food is quite tasty.)
3. Nós vamos visitar o México nas férias. (We are going to visit Mexico on vacation.)
4. Já estiveste no México? (Have you been to Mexico?)
5. O México é um lugar muito bonito. (Mexico is a very beautiful place.)