I think that names and terms of origin should act as a classification.
You can get carried away with that idea real fast, though. Which takes you back to square one of noncommunication.
For instance, the traits all those things mentioned above have in common is that they have cheese in them, and they are sauces. That constitutes a pretty huge class of things. Beer and stilton make a nice sauce, but it's not Mornay.
Classification is based on the idea that there's a system, an heirarchy that proceeds from the broadest to the narrowest, so that everybody can readily identify what's being discussed. Look at the way plants and animals are classified, for example. Using that as a model, one breakdown might be:
Order: Roux-based sauces.
Family: White roux-based sauces.
Genus: White roux-based sauces made with dairy.
Species: Cheese sauces.
What happens is that, as you follow the tree, each smaller branch has more things in common than not, until reaching the point where all the characteristics are the same.
In this rubric (and, in fact, in reality), Mornay is a variety that, while having all the broader characteristics in common with each other category, shares it's specific characteristics with no other product. It's classification, the way you are using the word, consists of a group with only one member.
On the other hand, there are many members of the group called "cheese sauces," many more in the group called "white roux-based sauces made with dairy," etc.
The way classic sauces are broken down does, indeed, follow such a pattern. It just doesn't use the same group names. We talk about mother sauces, and derivitive sauces, and derivitives of derivities. But in each case, a named sauce consists of specific ingredients handled in a specific way. Thus, we have White Roux sauces, of which there are two in the group (one made with dairy and one made with stock). The dairy-based mother sauce is called Bechamel, from which cheese sauces are derived, one of which is Mornay.
So, yeah, things fall into categories. But if we are to communicate there has to be a common language. Applying the name "Mornay" to a beer & Stilton sauce takes us away from the common language, and we no longer are communicating effectively.