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

No

While "fora" serves the same function as an adverb (meaning "outside" or "out") and as part of a preposition (meaning "out of") in both dialects, the answer is No because there is a difference in typical usage and meaning. In Brazilian Portuguese, "fora" is frequently used as a noun to denote a rejection, a snub, or a "blow" (e.g., dar um fora). In Continental Portuguese, this noun usage is not standard; a Portuguese person would more naturally use words like corte or rejeição to express the same idea.

Brazilian Portuguese Examples

1. Ela deu um fora nele na frente de todos. (She rejected him in front of everyone.)
2. Isso é fora de questão. (This is out of the question.)
3. Ele está fora do meu caminho. (He is out of my way.)
4. O time está fora do campeonato. (The team is out of the championship.)
5. Saia fora daqui! (Get out of here!)

Continental Portuguese Examples

1. Ela deu-lhe um corte à frente de todos. (She snubbed him in front of everyone.)
2. Isso é fora de questão. (This is out of the question.)
3. Ele está fora do meu caminho. (He is out of my way.)
4. A equipa está fora do campeonato. (The team is out of the championship.)
5. Sai daqui! (Get out of here!)