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

Yes

The word "banheira" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the vowels are typically more open and clearly articulated. In Continental Portuguese, unstressed vowels undergo significant reduction (they become shorter or nearly silent), which gives the word a more closed, consonant-heavy sound.

Brazilian Portuguese Examples

1. A banheira está cheia de água. (The bathtub is full of water.)
2. Eu amo tomar banho de banheira. (I love taking a bath in the tub.)
3. A banheira do banheiro está suja. (The bathroom tub is dirty.)
4. Coloque o sabonete na banheira. (Put the soap in the tub.)
5. A banheira é muito grande. (The bathtub is very large.)

Continental Portuguese Examples

1. A banheira está cheia de água. (The bathtub is full of water.)
2. Eu adoro tomar banho de banheira. (I love taking a bath in the tub.)
3. A banheira da casa de banho está suja. (The bathroom tub is dirty.)
4. Estou a limpar a banheira. (I am cleaning the tub.)
5. Põe o sabonete na banheira. (Put the soap in the tub.)