she has been like - it means "she" acts as if she is part of the family so far. the impression here is like that she "seems" to be a part of the family, but the acceptance is unclear
she's become like - "she" is accepted and treated like she is part of the family. this is more of a statement of fact, saying that she is no longer any stranger, but a part of the family, and not just acting like she is in one.
while they are similar, and probably, can be interchanged sometimes, the latter can be interpreted as more sincere.