It should be "How many recipes does he have left?" , so the recipes do not leave, but he has left some recipes.
That's the general sense, but I think it distorts the subject/object structure of the original French too much.
On the contrary, there is nothing wrong with the grammar of that sentence, although it is more common to insert a "for" or "to" between "left" and "him." (In this case, however, "left for him" is counted as wrong, so don't do that.)
I agree with SummerSun. You could perhaps get away with "How many recipes are left him?" in spoken English, but it's grammatically incorrect because "him" is an indirect object here and needs a preposition (e.g. for or to). It's certainly poor style for Duolingo to use it as the "textbook" response, as it obscures that "lui" translates as "to/for him" in this context.