I really don't understand your question. 0 is a number like all others and, like all other numbers, it is "unique"- it has some properties that other numbers do not. Yes, 0 times any number is 0, just as 2 times any number is even.
Abstractly, that property comes from the fact that 0 is the "additive identity" (x+ 0= x for any number) and the distributive law (a(b+ c)= ab+ bc).
x(y+ 0)= xy+ x0, by the distributive law.
x(y+ 0)= xy since 0 is the additive identity. Since xy+x0= x(y+ 0)= xy, we have xy+ x0= xy and, subtracting xy from both sides, x0= 0.