I'm not sure what you mean by "cross product" here, but there's nothing multiplicative involved. You just run both NFAs simultaneously. This gives you a number of states to track equal to the sum, not the product, of the two NFAs being intersected.
Also whatever you are describing is not an intersection per se as it’s not a permutation of the two possible combined states, it’s a construction for doing certain computations on intersected NFAs given its constituents. If each of two intersected NFA has three states the intersection can be up to 9 (the cardinality of the Cartesian product) states
If you loosen the definition of "NFA" that you're working with from requiring a set of final states to requiring a function from a set of states to {0, 1}, everything will still work exactly the same way, all of your theorems will still hold, but intersecting two NFAs will consist of adding one state and adjusting the accept function.
So not an NFA. That's fine, you can always define "extended NFAs", and some variants are practically useful. For example, most practical "regex" implementations have features like backreferences that are very convenient, but strictly more powerful than DFAs. You just have to be aware that you lose all the well-established theory around NFAs if you do something like that.
As I explicitly observed above:
>> everything will still work exactly the same way, all of your theorems will still hold
you don't lose any of the established theory by doing this.