Which kind of grammar is the following?
The simple answer would be that, having an infinite set of rules, this is not a grammar in the usual sense. Languages over infinite alphabets have been investigated, but usually using register automata rather than grammars.

That being said, you could treat the language as a limit (i.e. infinite union) of finitary languages defined by the "truncations" $G_i$ which you obtain by omitting all indices greater than $i$. I am unfortunately not familiar with research into such beasts.