In the various situations that I have observed, single mothers do not want their new male companions to be involved in the education of their child(ren). Living under the same roof as the child can become complicated when the child causes trouble in the accommodation.
If there is an argument between the stepfather and the stepson or stepdaughter, the child may reply, “you can’t tell me anything, you are not my father.” Then, the stepfather may talk to his girlfriend or new wife about the problem with her child, and she may respond, “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, and the relationship with the child can deteriorate. If the mother is on the side of the child, there is nothing the father can do. He will be in trouble with the two people he shares his life with.
For your information, I have never found a situation where a stepfather had legitimate authority over the children of his new wife or 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 as 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 may 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 trigger resentment against himself. The stepdaughter will hate him, and she will become his worst enemy.