It's honestly just a choice if you want to. I honestly thought it's like this at other parts of the world.
It's an Asian cultural thing. If you have watched korean dramas, you see that boyfriends/girlfriends often call their partner's parents eomoni and abeoji when they meet each other and same thing when you are married. Though, for males, there is also a different appellation.
I just said honestly twice (・・;)
If I had to guess, I would say it depends on the age of the child when the parent remarried and how involved the step-parent was in their life. If the parent remarried while the child was still young, and they grew up with their step-mom/step-dad being around them more than their biological mom or dad, then they might consider the step-parent to be on the same level as a parent. Or maybe they didn't get along well with their biological mom or dad, and are closer to the step-parent. Or, maybe the step-parent is strict and prefers to be referred to as "Mom" or "Dad" vs. "my step-mom" or "my step-dad" (I don't think most people would demand that of a kid, but I have heard of it before, or the opposite - "You don't *need* to call me Mom/Dad if you don't want to...")
My friend calls both her mom and her step-mom "Mom", and only clarifies "Well this is actually my step-mom" if someone asks her about it. But even on her phone, her step-mom's number is saved under "Mommy" and her biological mom is saved under "Mama". She is really close to her step-mom though. So yeah, I think it just depends on the person and what their circumstances are and how they get along with the step-parent.
I notice that in almost all type (yaoi, shounen-ai, or shoujo) of manga no matter if it Chinese, Japanese or Korean all the children refer to their parent spouse as their mother or father after their remarriage. Basically i am wondering if that is actually an Asian cultural thing or a manga thing. I refer to my father's wives as my father's wives. I have never once refer or thought of them as my mother and those that i know whose parents are remarried always refer to their parent spouse as "my step-dad" or "my step-mom" so i have been curious about why it was different in the mangas i read.