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

No

While "gato" primarily refers to the animal in both dialects, there are significant differences in typical usage, secondary meanings, and grammar. In Brazilian Portuguese, "gato" is widely used as slang for a handsome or attractive man, and it is the standard term for an illegal utility connection (such as electricity or water). In Continental Portuguese, while the word can be used to describe someone attractive, it is also used to describe someone sneaky or a trickster. Additionally, there is a grammatical difference in how the continuous tense is expressed (using the gerund in Brazil versus the "a + infinitive" construction in Portugal), and the pronunciation of the final "o" is much more closed in Portugal, often sounding like a "u."

Brazilian Portuguese Examples:

1. O gato está dormindo no sofá. (The cat is sleeping on the sofa.)
2. Aquele cara é muito gato! (That guy is very handsome!)
3. Ele fez um gato na rede elétrica. (He made an illegal connection in the electrical grid.)
4. Eu vi um gato preto na rua. (I saw a black cat in the street.)
5. O meu gato é o meu favorito. (My cat is my favorite.)

Continental Portuguese Examples:

1. O gato está a dormir no sofá. (The cat is sleeping on the sofa.)
2. Aquele rapaz é muito bonito. (That young man is very handsome.)
3. Ele fez uma ligação clandestina de eletricidade. (He made an illegal electricity connection.)
4. Eu vi um gato preto na rua. (I saw a black cat in the street.)
5. O meu gato é o meu preferido. (My cat is my favorite/preferred.)