I found this to be called: Not blood Related according to an article in TVTropes.
here are some examples:
In the anime Happy Lesson, Chitose's sisters aren't really related to him and are clearly invoking a Big Sister Instinct and Cool Big Sis.
KissXsis runs on this trope. Keita's twin step-sisters compete with each other for his affections, much to his dismay. His parents not only are aware of this, but actively encourage him to go for one of the girls, constantly reminding him that 'it's not by blood, so it's OK!'
In KissxSis, Ako and Riko literally loves their little brother Keita (related by marriage of their mother and keita's father) and they keep competing for his Love on almost every episode.
Freezing also has this, a 2nd year Pandora has to choose a 1st year limiter.. eventhough the limiter calls his pandora a sister (but not blood related) they perform an initiation "baptism" or also know as "first room visit"
Other examples: kanokon
Who started this? and what is its inspiration?