How common is the age play dynamic in age gap relationships?

I'm 21 years old, and I was wondering if it was very common to have an age play (ex: dad/son) dynamic in these relationships. I'm attracted to men of a large age range, including older men, but I'm not really into the dynamic, as I'm just more of a dominant person anyways; I could see myself as a caregiver but not as the one receiving care. There's nothing wrong with age play, I just don't feel comforted by or intested in being a "son".

I've had quite a few older men talk to me and hint at this (for example, referring to themselves as "daddy"). Is this something to expect if I date older men?