It's not lie if everybody accepts and supports it, merely a convention.
There's father-by-birth, and then father-by-adoption, and then father-by-convention.
There's father-by-birth, and then father-by-adoption, and then father-by-convention.
The daughter thinks the rental is her real father. That's highly likely to unravel sometime in her life time. Certainly at his funeral...he is using a fake name and personna.
The real father actually did disappear, after a history of abuse.
Not that someone has to be blamed, but it’s the shitty start of the story.
From there if the mother can fake it until her daughter is in a better situation, I think it’s positive.
Even if it doesn’t turn out that well, I am not sure it would much more devastating than having been bullied for all that time she was decieved, and it could a start for a discussion on how she deals with her father image.