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

Yes

The meaning, spelling, and grammatical function of "casaco" are identical in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the vowels are typically more open and articulated clearly. In Continental Portuguese, unstressed vowels (specifically the second "a") tend to be "reduced" or more closed, which can make the word sound shorter or more muffled to a Brazilian speaker.

Brazilian Portuguese Examples

1. Pegue o seu casaco, porque vai esfriar. (Grab your coat, because it's going to get cold.)
2. Eu adoro esse casaco jeans. (I love this denim jacket.)
3. Você viu onde eu deixei meu casaco? (Did you see where I left my coat?)
4. Esse casaco é muito estiloso. (This jacket is very stylish.)
5. Preciso lavar o meu casaco de inverno. (I need to wash my winter coat.)

Continental Portuguese Examples

1. Põe o casaco, que está a fazer frio. (Put on your coat, it's getting cold.)
2. Comprei um casaco muito giro ontem. (I bought a very nice coat yesterday.)
3. Onde é que pus o meu casaco? (Where did I put my coat?)
4. Este casaco é demasiado quente para este tempo. (This coat is too warm for this weather.)
5. Deves levar um casaco para a viagem. (You should take a coat for the trip.)