It doesn't actually matter whether or not the asker was assigned the problem as homework. The homework tag is for homework-like questions. These are questions whose purpose are not about resolving a conceptual difficulty, but instead about getting the right answer to a specific problem. Thus you can assign the homework tag without knowing if the problem actually is homework; homework-likeness is a property of the problem, not the asker.
I don't think people on this site care about whether or not you attend a university and I don't think they care about whether your question stems from a homework assignment or random curiosity. What they do care about is that your question addresses a conceptual issue that can be applied to many situations, so that the answer is useful to a wide audience and writing an answer is a good investment of time. The homework tag is supposed to indicate whether or not this is the case.
So I don't think we need a self-learner tag, because we don't care whether you go to a university or not, and we wouldn't need an "actual-homework" tag, because we don't care whether the problem is actually homework.
Here are some links to recent discussion about the homework tag for reference:
About the proposal to ban homework
What's the current status of the homework policy?