Internalizations is the process of describing a particular concept completely in terms of category theoretic language. Once that is done the same concept can be interpreted (internally, i.e., without need for any external ideas or concepts other than the category theoretic onces) in any category (provided the category has enough structure to for the description to be valid).
It is fair to say that subobjects are internalizations of subsets. A subobject classifier though is not the concept corresponding to characteristic functions. Instead, a subobject classifier is an internalization of the codomain of characteristic functions. It is what enables characteristic functions to exist (and characterize subobjects).