Are we "family"?

I'm trying to wrap my head around the idea of my kid having a sibling.

A little back info..my bf's ex has a son. She was pregnant when they were together so naturally my bf signed the birth certificate assuming the baby was his. He later learned the little boy wasn't(she cheated). He's gone through the necessary steps to ensure he doesn't have to pay her child support and even provided notarized letters to his ex so she can begin the process of changing her son's last name to hers(baby still has my bf's last name). But he hasn't gotten his name off the birth certificate because we've been told he absolutely needs a lawyer to do so and we don't have that kind of money.

Now me and him have been together for almost 4 years and we have a 8 month old daughter. My daughter and his ex's son have the same last name and legally on paper the same father. Does that mean they're step siblings?? It never dawned on me untill last night and I'm kinda in disbelief. I just need someone to either tell me that, yes, that's correct or that I'm just being crazy.

Granted he has not seen his ex or her son since they split but she insists to everyone and the little boy that my bf is his daddy. She claims she only has her son because my bf agreed to be a parent(she wanted to abort). But she failed to mentioned to my bf, at the time, that the baby wasn't his. She kept up the lie untill after she gave birth and it was sooooooooooo obvious the little boy wasn't my bf's biologically. But because he agreed she says the child is his responsibility.

I'm rambling but does that mean I should teach my daughter eventually that she has a step brother? Bc he's obviously being taught that my bf is his father. What happens if they meet one day and they both call him dad? Do I tell her it's a lie and his mom is psycho? Should I let them be siblings? His mom is the problem not him. I have nothing against the little boy and I'm sure he's a sweetheart. I'm so confused.