No, Action Surge, in 2014, gives you an additional Action. If you can make multiple attacks as part of a normal Attack Action, you would also be able to do so with the additional Action granted by Action Surge. So, it essentially doubles the number of attacks you can make.
You are correct, but it's only one additional action. The fighter already TOOK the multiple attack action, therefore they cannot use it again in the same round. It specifies only that you can take it when you take THE...not A...attack action.
Do we have something that clarifies that you can use the Action surge to give you that many attacks, or is this personal interpretations that it allows one to use Action Surge to take the Multiple Attacks ability as an Attack Action multiple times in a round?
My interpretation was always you can take multiple attacks and then you can use the Action surge for another attack (but not the multiple attack ability) as you have already used the multiple attack ability that round...and it is usage of THE Attack action, not An attack action.
Action Surge is on top of your regular action (which Multiple attack abilities fall under, whereas Action Surge is not one of the regular action you can take), and specifics it is one additional action...but does not specify.
It's an interesting interpretation, but it doubles the fighter's abilities quite immensely if you say it doubles their multiple attack action (which as I interpreted, you can only use once a round). If it were differently, then they should have defined that you can combine the two abilities within the descriptions (as they are both part of the default fighter) or clarified that you can combine these two things. They don't, and do not even indicate that these two items can be combined in such a way.
I know it's a popular interpretation...BUT...if this were so, then the multiple attack action should specify that you can do this. Otherwise it's basically doing what they (those who made 5e) said they wouldn't which is trying to make it a system mastery thing.
I'd like something official that specifies that it works this way, because as the fighter is written, it indicates nothing of the sort...and in fact sort of invalidates how the multiple attacks ability is written (because at the point you gain a second attack it should therefore specify that this actually means you get 4 attacks when using the action surge...not merely the two attacks).
If it were for a specific archetype, that would be one thing, but as they are BOTH default fighter abilities, I think it's more of a personal interpretation that's gotten popular to abuse, rather than official (though if you have an official Sage ruling or something like that, I'd love to see it).
PS: In answer to the question, if we are going to abuse Action Surge as suggested in this thread, the maximum number of attacks I've seen calculated is 1024. (PS: and no, this is not a joke. I've also seen someone hypothesize that they could at least double that with a specific weapon to 2048 attacks, though I haven't actually checked that one. These, of course, rely on abusing action surge as suggested, but they seem to work out if we are going that route).