Institute magic. An implant in Shaun's brain that allows Father to put him to sleep whenever he wants is one possibility. The Institute has some history with working with implants as shown by Kellogg. If someone is claiming to be my son, I want to see an unaltered DNA test to determine that a 60 year old man is related to me. If I go up to your house and claim to be your son, then you would obviously want far more proof than just my word. Instead Father uses some Institute magic by putting a 10 year old to sleep and claims that he is the Sole Survivor's son. Instead of asking for proof, the Sole Survivor idiotically believes it.
Although, the biggest problem showing that Father is most likely not the Sole Survivor's son is what kind of son would release the Sole Survivor into the wasteland? If Father was the Sole Survivor's son, then he would bring their parent to the Institute immediately instead of letting them wander the wasteland on their own with a very high chance of them dying due to Super Mutants, Raiders, Feral Ghouls, Death Claws, Radiation, Bloodbugs, Mole Rats, or robots. The possibility of someone from 2077 dying in the 2287 Wasteland is far too high.