Impulse response of a linear time-variant system
With the input-output relationship of a linear time-varying system given by

$$y(t)=\int_{-\infty}^{\infty}h(t,\tau)x(t-\tau)d\tau\tag{1}$$

it is straightforward to show that the kernel $h(t,\tau)$ (also referred to as impulse response or input delay spread function), must be unique.

Uniqueness means that there is no other kernel $g(t,\tau)\neq h(t,\tau)$ such that the output $y(t)$ is the same for all possible input signals $x(t)$.

Take the input signal $x(t)=\delta(t-t_0)$. The corresponding output signal is $y(t)=h(t,t-t_0)$, which, by definition, is different from the output signal $y(t)=g(t,t-t_0)$ obtained with the kernel $g(t,\tau)$.