There is nothing going on with these extended “symmetries”. If you give me a random thing to call a symmetry I can easily make up a charge formula for you.Kartik Prabhu

The universal structure is whatever structure (boundary, manifold/smoothness structure, certain fields like the metric) which are the same for all solutions you want to consider. The symmetries are the things which preserve this universal structure. A few examples:
1) Say you have a scalar field in Minkowski. What is the universal structure for all solutions? Well, it is the background Minkoowski spacetime; so the symmetries are Poincare. But if you look only at certain solutions, like rotationally-symmetric ones, then there is additional universal structure namely the rotational Killing field; so the symmetries are smaller than Poincare.
2) At null infinity, the universal structure is given by the usual conformal structure in the Penrose completion; the symmetries are BMS. If you choose to look at some smaller subset of asymptotically flat solutions, like stationary ones then the symmetries are reduced to Poincare instead of BMS.
3) At a finite null surface, you have much less universal structure, since a lot of quantities do not fall off and are not the same for all solutions. Consequently, you get more symmetries since you have less universal structure to preserve and so you can do more transformations. Again, if you look at a subclass of solutions like Killing horizons, there is more universal structure (the horizon Killing field) and the symmetries are also reduced.Kartik Prabhu

Those are not even continuous vector fields and don’t even preserve the manifold structure of the spacetime. So not sure why they are “symmetries”. If you are allowed those kinds of vector fields then might as well take a delta function vector field and call it “ultra soft hair”!Kartik Prabhu

I don’t know what “soft hair” means and what it has to do with supertranslations.
Null infinity is “less” symmetric than a horizon; it has a smaller symmetry group since null infinity has much more structure due to everything falling-off.
Supertranslations do not change the geometry, they are diffeomorphisms and just like any diffeomorphism they do not change the local geometry.