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

Yes

The meaning, spelling, and grammar of the word "poeira" are identical in both Brazilian and Continental Portuguese. The only difference lies in pronunciation: Brazilian Portuguese tends to have more open vowels and a more melodic, syllable-timed rhythm, whereas Continental Portuguese features more reduced, closed vowels (especially unstressed vowels) and a stress-timed rhythm.

Brazilian Portuguese Examples

1. A poeira cobriu todo o carro. (The dust covered the whole car.)
2. Preciso tirar a poeira da estante. (I need to remove the dust from the bookshelf.)
3. Tem muita poeira nesta sala. (There is a lot of dust in this room.)
4. A poeira levantou com o vento. (The dust rose with the wind.)
5. Cuidado para não respirar essa poeira. (Be careful not to breathe in that dust.)

Continental Portuguese Examples

1. A poeira cobriu o carro todo. (The dust covered the whole car.)
2. Tenho de limpar a poeira da estante. (I have to clean the dust from the bookshelf.)
3. Há muita poeira nesta sala. (There is a lot of dust in this room.)
4. A poeira levantou-se com o vento. (The dust rose with the wind.)
5. Cuidado para não respirares essa poeira. (Be careful not to breathe in that dust.)