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

Yes

The word "cenoura" is identical in both Brazilian and Continental Portuguese in terms of meaning, spelling, and grammatical gender (feminine). The only difference lies in pronunciation. In Brazil, the initial "e" is typically more open, and the rhythm of the word is more syllable-timed. In Portugal, the "e" is often more closed or even reduced to a near-silent vowel, and the prosody (the pattern of stress and rhythm) follows the more stress-timed nature of Continental Portuguese.

Brazilian Portuguese Examples

1. Eu gosto de comer cenoura no lanche. (I like to eat carrot in my snack.)
2. Você comprou cenoura para a sopa? (Did you buy carrot for the soup?)
3. A criança não quer comer a cenoura. (The child doesn't want to eat the carrot.)
4. Vou ralar uma cenoura para a salada. (I'm going to grate a carrot for the salad.)
5. Tem cenoura no bolo que você fez? (Is there carrot in the cake you made?)

Continental Portuguese Examples

1. Adoro comer cenoura crua. (I love eating raw carrot.)
2. Compraste cenoura para a sopa? (Did you buy carrot for the soup?)
3. A criança não quer comer a cenoura. (The child doesn't want to eat the carrot.)
4. Vou ralar uma cenoura para a salada. (I'm going to grate a carrot for the salad.)
5. Há cenoura no bolo que fizeste? (Is there carrot in the cake you made?)