"If someone would" - construction

We are talking about the subjunctive mood combined with the future tense of will, and the mood is applied independently to each part of the condition if X then Y. The first would makes Y tentative: it's dependent on the if X clause but indicates a future point in time after X. The second would makes X also tentative. You may think this redundant, but in fact the sense of the sentence change if X were definite and not tentative.

The writer could have written:

and you would say so, if the sultan allowed me to live another day.

This would be (sorry!) subtly different. This one indicates that the Sultan has not allowed the speaker to live another day, but considers what might have happened if that were not the case. This contrasts with the original example, in which the Sultan has not yet made his choice known, and so the situation being described is not exactly hypothetical: it might still come to pass that the Sultan will allow the speaker to live and therefore it might still come to pass that the addressee will "say so".

Note that if the construction had been would + had + verb-pp, you'd get a subjunctive mood about the past, similar to my first example above.

and you would say so, if the sultan would have allowed me to live another day

Here, the second would indicates a different past. This strongly, but not necessarily, indicates that the Sultan has not allowed the speaker to live. However, the would merely suggests, not indicates. Thus, it's a hypothetical past, not a definite one.

Consider:

If the train would have arrived 5 minutes ago, we would have caught it.

This does not necessarily indicate that the train didn't arrive 5 minutes ago. By contrast, this does:

If the train had arrived 5 minutes ago, we would have caught it.