Here is a counterpoint to some of the other responses. It's true that the abstract approach of Stuff, Structure, Property may seem sketchy (especially the passage to the core groupoid) but it is in fact surprisingly correct for some very broad classes of concrete categories with very rich notions of forgetfulness.
One such class (one that I am more familiar with) are the categories Mod(T) of models of a first-order theory T. There are three very distinct ways of forgetting things in Mod(T):
In the absence of evil and under other ideal conditions, the functorial characterizations of Properties, Structure, and Stuff translate to important results in model theory (various definability, interpolation, and consistency theorems). The translations are sometimes a little on the weak side, but I think that with adjustments to account for type information not captured by the theory alone, the translation can be made broader and even more precise.
To me, this is strong evidence that Stuff, Structure, Property is indeed the correct way to translate these notions of forgetfulness from the concrete to the abstract. While it's true that talking about forgetfulness without knowing what you're forgetting is nonsensical in when looking at particular instances, this approach provides a way of abstracting and even reasoning about forgetfulness in a completely general setting.
PS: Note that I am not a category theorist, I'm just a very impressed outsider. I suspect that category theorists have even stronger intuitions for Stuff, Structure, Property approach.