I think the best way to capture the intuitive necessity of 1+1=2 is to say that… Whatever is 2 is also 1+1, and whatever is 1+1 is also 2. So, to avoid mapping “+” to some empirical acting – instead the whole 1+1 is to be taken as a concept.

I think of that this way. If there is a pair of things in front of us (or pair of things imagined, or assumed, etc…) we can either a)put attention on the pair qua pair or b)put attention on each of individuals as another to each-other. I think that a) corresponds to something being determined as 2, and b) corresponds to something being determined as 1+1. While there is the pair in front of me, I can switch my determination of it between a) and b) in similar way as it happens with the Necker cube. Somewhere there I think the necessity is “seen” (or intuited/comprehended).

So, I think that what happens in 1+1=2 is not identifying the left and right side of it as something separate, but as necessity if something determined as 2, it will also be possible to determine it as 1+1 (and the other way around).

BTW, I don’t think this is truth based on the definitions on the terms, or some formalism. I think a person can know what ‘1’ is and what ‘2’ is, and not know if 1+1=2. (Of course that doesn’t mean that those symbols can not be used in a formalism, that would map in some way to the intuitive understanding, which I think is the case with the idea of axiomatization of math)