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

Yes

The word "assim" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only differences are phonetic. In Brazilian Portuguese, the vowels are typically more open and the rhythm is more syllable-timed. In Continental Portuguese, unstressed vowels (such as the "a" in "assim") are often more closed or reduced, and the cadence of the speech is more stress-timed, which can change the perceived weight of the word within a sentence.

Brazilian Portuguese Examples

1. Não fala assim comigo! (Don't talk to me like that!)
2. Eu gosto de tudo assim, bem simples. (I like everything like this, very simple.)
3. Assim fica muito mais fácil de fazer. (This way it's much easier to do.)
4. É assim que a gente faz as coisas por aqui. (This is how we do things around here.)
5. Tudo aconteceu assim, do nada. (Everything happened like this, out of nowhere.)

Continental Portuguese Examples

1. Não me fales assim! (Don't talk to me like that!)
2. Gosto de tudo assim, muito simples. (I like everything like this, very simple.)
3. Assim fica muito mais fácil de fazer. (This way it's much easier to do.)
4. É assim que se faz. (This is how it is done.)
5. Aconteceu tudo assim, de repente. (Everything happened like this, suddenly.)