Is there any connection between frequency response of a system and spectrum of a signal?
As mentioned in my comment, look at the Wold Decomposition Theorem :

https://en.wikipedia.org/wiki/Wold%27s_theorem

but actually showing how it relates to your question,

http://www.phdeconomics.sssup.it/documents/Lesson11.pdf.

In simpler terms, your AR system can be represented by a MA system (possibly infinite order) and as the second reference shows, an ARMA system.  The process has more than one representation so there isn't a one to one correspondence between the time series and some system that generated it. 

So the answer is yes, there is a correspondence and often the system itself, is of interest, but the time series and system are not uniquely equivalent. 

another way of thinking about it is that $H(\omega)$ has a phase and magnitude response that gives $|H(\omega)|^2$ but there are other possible phase responses that results in the same magnitude squared.