Does XNOR of three variable equals XOR of same three variables
XOR is addition modulo 2, and XNOR computes the sum modulo 2 of its inputs and 1. Since
$$ (x+y+1)+z+1 \equiv x+y+z \pmod{2}, $$
We see that XORing three variables is the same as XNORing them. The same holds for any odd number of variables. When XNORing an even number of variables, you get the negation of their XOR.

Nothing of this sort happens for AND or OR. There isn't any particular reason. You should think of it the other way around: XOR satisfies this surprising property, but other gates do not. XOR is special. It has absolutely nothing to do with AND and OR being "basic gates", whatever that means (mathematically, probably not much).