Step daughter 8 and step son 5?
I certainly would not jump to abuse just because they like it better by you than with their real mother, lol.
If it really concerns you then look into it a little but I highly doubt thats the case here.
Its really sweet that they call you mom by choice, but I think thats a little messed up, dont you? They have a mother and its an insult.
They are children. You said their mom is strict. Those two things dont mix well. They are also probably more emotional than rational at this age. Which is more likely than not, the reason they get upset when they leave your home.
You and their mother also shouldnt have conflicting rules for the home. It greatly confuses the child(ren) and is probably also making them behave the way they are. I know its hard being a step mom and finding out your exact role in their lives but you and the mother really should be on the same team lest you unintentionally turn them away from their mother.
Im sorry if any of this was hard to hear but you needed to hear it judging by your post.