To get the behavior you want, your class needs to overload numArgumentsFromSubscript. One example implementation of this method would look like the following. The rest of your class would remain the same.
function n = numArgumentsFromSubscript(obj, S, ctx)
n = builtin('numArgumentsFromSubscript', obj, S, ctx);
end
The point of numArgumentsFromSubscript is to tell MATLAB how many outputs the particular instance of your class returns when indexed in a certain way.
When I add this method to your class, the new class works the way you expect:
>> enhancedSubsrefExample = EnhancedSubsrefExample([EnhancedSubsrefExample(1), EnhancedSubsrefExample(2)]);
>> [enhancedSubsrefExample.Property.Property] % Returns [1, 2]
It's also worth nothing that in recent releases, you can use modular indexing mixins such as RedefinesParen and RedefinesDot in place of overloading subsref and subsasgn. If you are able to use modular indexing, it will generally result in better performance when your class overloads indexing.
