There are always exceptions. I don't tend to rely on their occurance.
I'm not primarily talking about a mans authority over the step child, but over his wife. Too often he becomes the third wheel in the relationship. There is this tendency among wives to put the children first in the relationship (I'm talking about relationships where the biological parents are together) and to use them as leverage against the authority of the father. She's primarily there as a mother, not as a helpmeet. That is backwards. These sorts of things lead to trouble in the empty nest phase when they suddenly find they've "grown apart". Or divorces soon as the youngest is out of diapers. Single mothers are that same affect on steroids, and baked into the cake.
But now that you bring it up, it does really seem odd to me to have children in my house, eating my bread, but completely not under my authority. I won't say its wrong, I have no experience for how to navigate that situation. Just feels odd.
Where there is always that option. Not sure how much I should count on it though.
Ha! I've lost count of the number of single mothers I've seen who come right out front saying "my kid will always come first" as if its the most important thing you need to know about her. It's so common as to be a cliche.