In the various situations that I have observed, the single mother did not want her new male companion to be involved in the education of her child(ren). Living under the same roof of this child gets complicated when the child causes trouble in the accommodation.
If there is an argument between the stepfather and the stepson or stepdaughter, he/she can reply: “you can’t tell me anything, you are not my father”. Then, the stepfather may talk to his girlfriend / new wife and tell her about the problem with her child. She may answer: “You are not his biological father, you should not be involved in this issue”.
The relationship may be perfect at the beginning but at any point, everything can change in time and the relationship with the child can deteriorate at any point. If then, the mother is on the side of the child, there is nothing the father can do. He will be in trouble with two persons he shares his life with.
For your information, I have never found a situation where a stepfather had a legitimated authority over the children of his new wife / girlfriend. I have never found a situation where the mother said to the children: “now my new companion will have as much authority as your biological dad”.
There is an additional issue. The stepfather may not have the same views on education than the biological father. For example, we know fathers who tell their children that they can go to sleep whenever they want and a stepfather who will ask the children to go to sleep at 9 PM. This creates tensions that no-one can fix. During an argument before sleeping, the stepfather can tell his stepdaughter that she will have to do what he wants her to do when she stays under his roof. Doing so, he will just trigger a resentment against himself. The stepdaughter will hate him and she will become his best enemy.