I had this problem, not as soon as 4 months though. I have been slated to no end for it.
My DD started asking to call my (now) husband dad, especially when SDS's were visiting and obviously calling him dad. We told her no, she has a dad, never once encouraged it. Generally DD ignored us and would constantly slip into calling him dad when the boys were here, we would always correct her but it would make her angry.
Then we had a DS together so I also started referring to my him as dad. After we got married it became worse, she started calling him step-dad and saying "well he isn't, isn't he? so why can't I call him that?" After the birth of our 2nd DS, and our first DS was old enough to be calling him dad, my daughter would fly into a rage about being the only child of 5 not calling him dad.
After seeking professional advise I was told that it was extremely important for her to feel included and not like an outsider in her own home/family. Even I refer to him as dad... he's not my dad either but I call him dad all the time, he calls me mum too, its what parents with children do! Arguing with her was doing more damage and he is a father figure, he is her step-DAD. I backed down and stopped fighting with her / correcting her anymore.
I've had no end of abuse over it and so has DD, from her paternal relatives, poor girl, they tell her she is "betraying her dad". Not long ago I found out that dad had been trying to get DD to call one of his GF's mum, boy did he make a rod for his own back, DD's blasted him for bullying her over calling her step-dad dad when he'd tried to get her to call his GF mum.
Truly, my life feels like a bad episode of EastEnders :(