How to quantify amplitude of oscillations in marginally stable systems?
Is amplitude of oscillation a (only) function of initial condition of the system?

No.  For a system with a transfer function of the form $H(s) = \frac{\cdots}{s^2 + \omega_0^2}$, the amplitude of oscillation will increase any time it is excited by a signal that has a component at $x(t) = \cos \omega_0 t + \phi$.  Even if you're not exciting it intentionally, random noise will excite that pole.  Basically, the output will be of the form $a(t) \cos \omega_0 t + b(t) \sin \omega_0 t$, where $a(t)$ and $b(t)$ will be Wiener (random-walk) processes.  Such processes have variances that tend to infinity as $t \to \infty$.

How to intuitively quantify amplitude of the oscillations for marginally stable system?

&quot;Big and growing, boss, and I don't think they'll get smaller!&quot;
Possibly by how fast they grow.  But grow they will.

In some applications, oscillations with small amplitude might be acceptable.

If you have a system that's exhibiting persistently small oscillations, then you're seeing the signs of a nonlinear phenomenon called a &quot;limit cycle&quot;.  Basically, there's some oscillator in there that's either inherently small-valued (like the oscillation you may see around the least significant bit of a DAC or ADC), or there's a big oscillation that's mostly not getting to the output.  Either way, if it persists at some amplitude, there's some nonlinear process that's keeping it that way.