What kind of learning problem is this?

Say I have $n$ multi-class classification problems $p_1$, ..., $p_n$. Each of these have their own training data. While they are all distinct problems, there may be similarities in their data (which are in my case images), e.g. the data for class $p_1^{_B}$ of problem $p_1$ may be similar in some way to the data for class $p_5^{_F}$ of problem $p_5$. Classically, each classifier tries to seperate each of its training classes from each of its other training classes, but doesn't look at the other classifiers and their training classes. If we instead train the classifiers jointly somehow, can the relationship between the classes (from different problems) be made more explicit? For example, maybe any image classified as $p_3^{_C}$ would also be classified as $p_7^{_A}$ and vice versa, or any image classified as $p_2^{_B}$ or $p_2^{_C}$ would be classified as $p_5^{_F}$, with some degree of accuracy?

Possibly related things that I have explored but don't seem to quite fit (simplified to refer to just two problems $p_1$ and $p_2$ but in principle generalizable to $p_n$):

1. Transfer-learning seems to be concerned with how to use the knowledge gained from solving (say) $p_1$ to better solve (say) $p_2$. Optimizing the accuracy of any of the classifiers is not myr goal.
2. Multi-task learning (MTL) seems to be concerned with how to use the knowledge gained from solving both $p_1$ and $p_2$ together, to improve the performance on both. Again, optimizing the accuracy of any of the classifiers is not my goal. But, the core intuition of MTL is that it may be beneficial to exploit the hidden relationships between the two (or more) tasks, and this is essentially (it seems to me) the same premise as in my task.
3. Multi-label learning seems like a reduced form of my task, arising if each of the classification problems $p$ are binary instead of multi-class. What I find interesting is that label graphs seem to have the same general goals as in my task.

My question is: assuming that my task has already been described and studied in the vast machine learning literature, what is the name for it? This would help me find relevant papers and potentially software, so I don't have to reinvent the wheel. Finally, I am a machine learning beginner, so this question could well be fundamentally misguided. If so, please correct.