When applying the Separation of variables technique for homogeneous linear constant coefficient PDEs, we get equations of the form
$X''(x)/X(x) = T''(t)/T(t) = \mu $
Which arise from the trial solution $ \ u(x,t)=X(x)T(t) \ $, substituted in a 2nd order PDE in $u(x,t)$.
Why does the Separation constant $\mu \ $ necessarily exist, and why is it unique for all $x$ and $t$ ?
The separation constant seems to be the intersection of two functions, so in practice, it might not exist, or could have several values depending on the functions involved?
It is impossible that a function $f(x)$ be equal to a function $g(t)$ for any $x$ and $t$ except if both functions are equal to a same constant, say $f(x)=g(t)=\mu$.
Let $f(x)=X''(x)/X(x)$ and $g(x)=T''(t)/T(t)$, thus $$f(x)=g(t)=\mu=X''(x)/X(x)=T''(t)/T(t)$$
Insofar $f(x)$ and $g(x)$ exist, $\mu$ exists.
But $\mu$ is not unique. $f(x)$ and $g(x)$ are necessarily both the same constant function. But it is totally possible that the constant $\mu$ be any constant.
For each arbitrary chosen constant $\mu$ solving the equations $X''(x)/X(x)=\mu$ and $T''(t)/T(t)=\mu$ for $X(t)$ and $T(t)$ leads to a different solution $u(x,t)=X(x)T(t)$. Each one is solution of the PDE. And any linear combination of those elementary solutions is also solution of the PDE.