I've seen historical Chinese blood tests in fiction before, but never really got how they worked.
It's based on hemolytic reaction from mixing blood. You take some blood from the two people (father and son in this case) and mix it together.
The principle is that if they're closely related, nothing will happen, but if they're not related, the blood will react and destroy each other.
This is due to blood type: If two two are the same blood type, there's no reaction, but different blood types destroy each other (which is why it's really bad to get a transfusion of the wrong blood type).
The test isn't reliable at all though, since a child won't necessarily have the same blood type as their parent, and also a lot of people have the same blood type.