While discussing eigen functions of LTI system, why we ignore the natural frequency term in response?
The eigenvalues are the roots of the characteristic equation, so \$s\$ needs a value in your first \$e^{st}\$. Generally, if \$s=-\sigma_1\pm j\omega _1\$, is a complex conjugate pair, then the corresponding eigenvalues will be \$e^{-\sigma_1 t}(cos(\omega _1 t)\pm jsin(\omega_1 t))\$, and \$\omega _1\$ will be the natural frequency.

In your second \$e^{st}\$, \$s=j\omega\$ represent a forcing sinusoid at an arbitrary frequency, \$\omega\$.

If there are no complex or imaginary eigenvalues, there is no natural frequency, just 1st order real exponential terms.