THANK YOU!! And thanks to all who took my side.
For those of you who took my 15 yo son’s side over mine, you can all go straight to hell.
I kid, I kid!
It’s not entirely clear to me. In this thread, are we talking about the more traditional usage of “all” to mean “every member of a particular set”? Or are we talking about the more slangy usage of “all” to mean “a lot, but clearly not all,” as in “I have all the feels” or “Right after I got over the flu, I came down with appendicitis; I’ve had all the illnesses!”
This may have an effect on the correct answer.
Yes, this. It ain’t rocket surgery. The very basic question is, how many in a particular set is required, once every member of that set is accounted for, to refer to that particular set as “all”? Does it have to be a minimum of 3, or can it be 2 (or even fewer, as @glowacks postulates)?
I’ve heard “all three of them” before but never “all two of them.”
Hi Opal!
I agree that all two of them wouldn’t be standard. But it’s not the construction solost posted.
Thinking about it, in slangy speech between friends, all two of them might be a proper snarky statement. “He’s always bragging about how many celebrities he knows. All two of them.”
I would say it like this to your kid: “All” implies 3 or more, but does not outright require it.
In your case, I would definitely say “all the cats” for the reason @pulykamell indicates. You’re emphasizing the fact that there are no other cats. Plus you’re clearly speaking in a rather informal register.
It would be like me telling your kid that were wrong to not include a period at the end of a text.
I voted for two, but really, I would have liked to have seen an option for just one, too. To me, if you have even one cat, you can say “all of the cats.”
Well, I can’t change the poll now, but how about we just take the ‘2 or more’ option to mean any amount, down to one or even 0? With 22 votes in I’m behind 41-59%. I was hoping to triumphantly prove my know-it-all son wrong, and y’all are letting me down! 
Many Germanic languages retain the dual (as in singular - dual - plural) in some contexts. This is one of those contexts. If you want to just ignore decent grammar you can say “You have all the cats” even when there is just one or two cats, but both those usages are an insult to decent grammar and our ancestors.
Wow, naita, don’t hold back, tell me how you really feel. Ouch! I consider myself a bit of a stickler when it comes to proper grammar and usage, so this hurts, not gonna lie. Sorry, ancestors!
Aaaand, I’m still losing in the poll with 28 votes in. It’s looking like I may have to concede this round to Mr. Know-It-All. Curses!
Or even “All one of them.” There’s always a snark exception.
Right. Context and tone always matter. There is only one firm rule for usage, especially for casual spoken usage. The only firm rule is that one can never declare that a firm rule from formal written usage must be applied.
I suggest making lemonade from these lemons by taking pride in your son’s excellent grasp of pedantry and prescriptivist grammar. 
I think if you have a set containing all elements, you can say “all”, even if the set contains only one element.
But I’m weird like that.
“All” - “the whole number or amount of or every one of a class”. If “everyone of a class” amounts to only 2, then 2 is “all”.
Yes, and, frankly, I can’t believe this poll is 50-50. For the prescriptivist pedants, there is nothing in the dictionary definition of “all” that says it has to be at least three items, or even at least two, or even at least one. It’s however many is in the set. If I had two eggs left in a carton of eggs, and then I go to the fridge and they’re gone, I’d ask my wife “did you eat all the eggs” or “did you eat the rest of the eggs” without a second thought. In that context, “did you eat both the eggs” sounds weird to me. And I’ve certainly heard “all” in reference to one item in speech, usually in the context of placing emphasis one the fact, like I said before, that the one item represents ALL the items in existence. It works both prescriptively and descriptively, although descriptively there is a preference to express “all” with larger quantities. It’s not required, though, and preferred in certain contexts to make a point, especially as in the OP.
If I had the only apple, and someone asked, “Do you have all the apples?” the answer would be yes. Grammatically, “all” doesn’t need a quantity, but it certainly carries the implication of quantity. I voted for three or more because that’s how I would use it, but in colloquial speech I’m not fussy about it.
There is some subtext to the word all.
“In 2000, I booked a trip to Disney, and brought all my ex-wives and all my children to enjoy the Magic Kingdom with me.” I had never been married, had no kids and went with my cousin, but it is grammatically correct.
It isn’t required to have more than 2 items to be grammatically correct, but the use of all with 2 or fewer items can definitely sound stilted or give a wholly incorrect impression of what you’re trying to say.
But not in the OP’s case. I argue that it is absolutely appropriate in that case for emphasis and neither non-colloquial nor grammatically incorrect.
In a slightly different case, I find it quite commonly used for two or even one. Like my wife the other day asked me “did you drink all the soda?” and I replied "yes. (There was only one can left, that was the only one I drank, but I drank it. Perhaps it would better have been phrased as “did you drink the rest of the soda?” but that also may have an implication of multiple sodas to people. I come across “all” in this usage for “unknown quantify of items in a set” reasonably regularly.)