It does have to do with utilitarianism — if you change the mugger to harming Bentham, the situation is different. In that situation, many other reasonable moral theories would agree with utilitarianism.
In the original situation, where the mugger is harming themselves, the critique is that utilitarianists are required to treat their own interests as exactly the same as other people’s interests. It doesn’t matter if someone is harming themselves in order to provoke some action from you; if your action prevents that harm, you are obligated to do that action (even if you suffer because of it).
In the original situation, where the mugger is harming themselves, the critique is that utilitarianists are required to treat their own interests as exactly the same as other people’s interests. It doesn’t matter if someone is harming themselves in order to provoke some action from you; if your action prevents that harm, you are obligated to do that action (even if you suffer because of it).