This is actually possible to answer, by looking at how we use both "times two" ("times three", "times four", etc.) as a standalone phrase to multiply whatever comes before it and "two times" ("three times", "four times", etc.) as a standalone phrase to multiply whatever comes after it.
Here are a few cites from the Corpus of Contemporary American English:
*
Do you know what it's like to have triplets? It's just everything times three: three pairs of grubby little hands, smearing food all over your brand new Chihuahua outfits; three pairs of feet in their weird, soft little shoes; three pairs of eyes, brown, like my husband's.
-
Iran is a much larger country, geographically, than Iraq. It has three times the population
-
There was no way she was going to have to pay $333 an hour, times three, for having sex with this guy.
-
At stake are 518 delegates, more than three times the number awarded so far in the unpredictable competition
-
New paint. New furniture. All those special touches for a new baby times three.
-
a mostly Athenian army crushed a veteran Persian force three times its size
-
— The 88 keys were taking over the family.
— That is a true statement. I mean, literally, because it wasn't 88 keys. It was 88 times two, times three, times four, and eventually times five.
— Five grand pianos in the house?
— Yes.
-
LED can run two to three times the cost of conventional lighting sources
-
The next month you write three $100 checks and put each into an envelope that asks only for $50, and now you feel as if you had real money, savings, like you've spent $50 and saved $50, as if you have $50 to spend, and that's times three.
-
Mercury times three
In three flybys, MESSENGER snaps portraits of portions of Mercury[.]
-
And they suffered nearly three times the number of wounded
-
Nefertiti Times Three
-
I saw a knife, a guy that was probably three times the size of my dad
-
After "Sleep tight, sweetie" times three, Jonah and I were rarely finished being parents.
-
FUNAI patrols 54,000 square miles of remote territory, an area more than three times the size of Switzerland
-
I could sense what other people were feeling three times as deeply as other kids, and my reactions would be times three. [Lo and behold — both variants in the same sentence!]
There are many more, but you get the drift.
If "A times B" had only one possible parse tree, only one of the two variants would be productive, and so half the above quotes would be ungrammatical, and indeed nonexistent. But they are all there and they are all fine, which goes to show that multiplication is commutative not only in maths, but also in the English language.
TL;DR: Sometimes, in "A times B", the "times B" part is considered to be a single unit, other times it's the "A times". Both readings are possible and productive, and often enough, you can even use them completely interchangeably without restructuring anything. You can patrol an area three times the size of Switzerland, and you can patrol an area the size of Switzerland times three.
So a 2×3 is a 2+2+2 is a 3+3.
Edit: It is important to note that this is not universally true of all languages. For example, in Russian, "three times two" can only be parsed as "[three times] two"; the parsing "three [times two]" is impossible. So wondering how English handles this is indeed a valid question, and indeed one about English rather than about how the human brain generally works.
