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

Yes

The word "gostam" is identical in spelling, meaning, and grammar in both Brazilian and Continental Portuguese. The difference is strictly phonetic. In Brazilian Portuguese, the nasalized vowels in the suffix "-am" are typically pronounced with more clarity and a more open sound. In Continental Portuguese, there is a much stronger tendency toward vowel reduction, meaning the unstressed syllables in "-am" are often shortened, compressed, or nearly "swallowed" in speech.

Brazilian Portuguese

1. Eles gostam de viajar nas férias. (They like to travel during the holidays.)
2. Vocês gostam de filmes de ação? (Do you all like action movies?)
3. Eles gostam muito do ônibus. (They like the bus very much.)
4. Eles gostam de usar o celular. (They like to use the cell phone.)
5. Eles gostam muito do cachorro. (They like the dog very much.)

Portuguese

1. Eles gostam de viajar nas férias. (They like to travel during the holidays.)
2. Vocês gostam de filmes de ação? (Do you all like action movies?)
3. Eles gostam muito do autocarro. (They like the bus very much.)
4. Eles gostam de usar o telemóvel. (They like to use the cell phone.)
5. Eles gostam muito do cão. (They like the dog very much.)