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

Yes

The spelling, meaning, and grammatical category of "garagem" are identical in both Brazilian and Continental Portuguese. The only difference is in pronunciation:

• Brazilian Portuguese: The "r" is often pronounced as a guttural or aspirated sound (similar to the "h" in "house"), and the final syllable features a distinct, nasalized "e" ([ẽ]).
• Continental Portuguese: The "r" is typically an alveolar tap or trill, and the vowels are generally more closed and less nasalized than in most Brazilian dialects.

Brazilian Portuguese Usage

1. Deixe o carro na garagem. (Leave the car in the garage.)
2. A garagem está muito bagunçada. (The garage is very messy.)
3. Eu estaciono na garagem de casa. (I park in my house's garage.)
4. Tem muita coisa guardada na garagem. (There is a lot of stuff stored in the garage.)
5. Você viu onde eu deixei a chave da garagem? (Did you see where I left the garage key?)

Continental Portuguese Usage

1. Deixa o carro na garagem. (Leave the car in the garage.)
2. A garagem está muito desarrumada. (The garage is very messy.)
3. Estaciono na garagem de casa. (I park in my house's garage.)
4. Há muita coisa guardada na garagem. (There is a lot of stuff stored in the garage.)
5. Viste onde deixei a chave da garagem? (Did you see where I left the garage key?)