There are ill-posed summations that we can assign values to, take for concreteness, $$ S = \sum_{k=0}^\infty k $$ to which we can assign $-1/12$ by several methods. Is there a fundamental and rigorous reason why these methods have to agree on the same value?

Perhaps the most powerful means we have available is the zeta function regularization, where we attach $\zeta(-1)$ (Riemann zeta function), which formally represents $S$, to the summation. If we have another analytic function that formally represents $S$ and analytically continues to assign a value to $S$, must the value always be the same, $-1/12$?

More concretely, suppose I have a function $\xi(s) = \sum_{k=0}^\infty f_s(k)$ such that $f_1(k) = k$ and $f_s(k)$ is a "nice" function (for the case of zeta function, $f_s(k)=k^{-s}$ is the exponential function in $s$). Suppose that $\xi(s)$ can be analytically continued to a meromorphic function with a value at $s=-1$. Must it be the case that $\xi(-1)=-1/12$?

More generally, what can be said about other divergent sums? If there is an analytic continuation that defines the value of such sums, must it be unique? Where can I look up known uniqueness results and techniques?