It took us thousands of years to properly define real numbers. High school students can live without a perfect explanation, or we can just teach limits before college since they are the fundamental concept if calculus.
Probably you were taught how to multiply irrational by the property of powers (a^b * c^b = (a*c)^b etc.).
You were not taught a grand unifying theory of multiplication, you were taught how to manipulate operations to turn them into more useful operations.
Teaching these laws also prepares you for when a and b are just symbolic reals with no structure and those laws are the only thing you can use to manipulate them.
You can't teach "the truth" (whatever you hold that to be). It would set back education instead of advancing it. In this case too, perfect is the enemy of good.
Your example is quite bad because sqrt(2)*sqrt(2) = sqrt(2*2) = sqrt(4) = 2. So repeated addition works fine. Let's focus instead on pi*pi. The way calculators do this is precisely as some type of limiting sequence depending on how much precision you want. Because, one cannot "calculate" pi*pi exactly because it is irrational. So, you have 3*3, then 3.1*3.1, then 3.14*3.14, etc. which are all repeated additions with some division (e.g. 314*314/(100*100)). In reality, when multiplying two irrational numbers, we just use enough decimal points for floating point precision and then chop off any potentially erroneous digits after the multiplication.*
As others have pointed out, repeated addition as multiplication readily extends to rational numbers, then to irrational numbers as limits of rational sequences. This is exactly the progression that is taught in Rudin's analysis book and the way to construct the real numbers. At no point in time do you need to backtrack on repeated addition but you need to introduce new concepts division and limits. This is exactly teaching F=ma and then introducing relativity and quantum as the students gain more depth and break past the classical setting.