How to use MATLAB to observe the effect of multiple damping factors?
It's called system identification. This is a good starting point:
http://mocha-java.uccs.edu/ECE5560/ECE5560-Notes04.pdf



You could then relate the estimated transfer function back and find zeta.
(if you just want to find zeta now, you can already do that with the numbers you have, and you don't need system identification).

To find an estimated transfer function, you need to know the input and outputs of a system, you can then find the relationship between the inputs and outputs which is the transfer function.

I am fairly sure that this can't be done accurately real-time. To know the frequency response of a system, you need to have enough of the frequency space sampled. But you can do this a short time after the system is settled. (or you could calculate zeta in real time, it might not make sense until the system is settled).

There are other tools and code already written to estimate transfer functions, I hope this gives you a good starting point.