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

Yes

The spelling, meaning, and grammatical function of "apartamento" are identical in both Brazilian and Continental Portuguese. The only difference is in pronunciation. Brazilian Portuguese features more open vowels and a more melodic, "vowel-heavy" rhythm. In contrast, Continental Portuguese is characterized by more closed vowels and a "staccato" rhythm, where unstressed vowels are often reduced or nearly silent.

Brazilian Portuguese

1. Eu moro num apartamento pequeno. (I live in a small apartment.)
2. Esse apartamento é muito caro. (This apartment is very expensive.)
3. Estou procurando um apartamento para alugar. (I am looking for an apartment to rent.)
4. O apartamento dele é bem grande. (His apartment is quite big.)
5. Meu apartamento fica perto do metrô. (My apartment is near the subway.)

Continental Portuguese

1. Eu moro num apartamento pequeno. (I live in a small apartment.)
2. Este apartamento é muito caro. (This apartment is very expensive.)
3. Estou à procura de um apartamento para arrendar. (I am looking for an apartment to rent.)
4. O apartamento dele é bastante grande. (His apartment is quite big.)
5. O meu apartamento fica perto do metro. (My apartment is near the metro/subway.)