We could figure out that our symbolic manipulation is inconsistent with the axioms based on the quirk you consider.

There are more axioms needed to define Peano Arithmetic. Taking axioms 7 and 8 from Wikipedia, translated into your notation:

\a. Sa != 0 \ab. Sa = Sb -> a = b

(I also use the symmetry and transitivity of equality.)

Note, from the axioms you stated:

\a. S0 + a = 0 + Sa = Sa

So, axiom 8 can be restated as:

\ab. S0 + a = S0 + b -> a = b

So, starting with your result:

SSS0 = SS0 + SS0 = S0 + SSS0

But also,

S0 + SS0 = SSS0 = S0 + SSS0

So, by the restatement of Axiom 8:

SS0 = SSS0

And then using the original form of Axiom 8 twice:

S0 = SS0 0 = S0

We have a contradiction of Axiom 7.

Thus, it is proven that our symbol manipulation does not follow the Peano Axioms. This does not invalidate the Peano Axioms. It simply means that a given physical system does not follow them. Of course, it would be difficult for people living in an alternate universe where symbols really behaved this way to notice the distinction. And they likely would not have to, if all objects behaved that way; they would figure out some other math to represent their situation. And at some point, mathematicians in their ivory towers would develop this weird math that is really hard to write down and has no known application in the real world (both properties bringing great joy to these academicians).