What's the meaning of the continuity in spectrum analysis?
Let $v(x, k)$ be an orthonormal set of basis functions such that
$$
\int_{-\infty}^{+\infty} v(x,a) v^*(x,b) dx = \delta_{a,b}
$$
where $\delta_{a,b}$ s the Kronecker delta.

I believe your question boils down to: for the Fourier Transform, we have
$$
v(x,a) = e^{jax}
$$
and we interpret $a$ as the frequency and $x$ as time; is there anything special about this orthonormal basis because we can interpret $a$ as a "distance" between these functions?

An example of another set of functions that are orthonormal (but over a finite interval) are the Legendre Polynomials:
$$
v(x,a) = P_a(x), \mbox{where} -1\le x \le +1
$$
where now $a$ is the order (degree) of the polynomial.

For these (and other polynomial bases), the "distance" is now the difference in polynomial degree.

Whether this means such functions are "less" special than the Fourier bases is a matter of opinion.