You can assign anything to anything, but it doesn't mean it's true. Here the sum is defined for complex numbers with real part larger than 1 and it matches the Riemman zeta function on that domain. Outside that domain the sum is no longer defined, but its extension (Riemman zeta function) is. And that extension's value at -1 is -1/12.

The extension being the most natural in no way means our sum needs to have the same values. Maybe our formal definition of a sum is just not the most natural one.

An example: I define a function f explicitly, by f(x) = sin(x) if 0 < x < 1, otherwise the function is not defined. The most natural extension of f from (0,1) to IR is sin. But that doesn't mean f(100) = sin(100). f(100) is not defined by our definition.

If you prove that the sum of all natural numbers is -1/12 (by our standard definitions), you prove that math is contradictory.