Z-parameters actually represent impedance, but this is not the same as characteristic impedance of a transmission line.
Characteristic impedance of a trace assumes that the trace has the same cross-section everywhere and is infinitely long. The characteristic impedance then is the same at any point along the transmission line.
Z-parameters are measured at specific points (ports) of a finite network. At a port, the impedance is not the characteristic impedance of a trace, but it is the actual impedance measured at that point in the network. Because the network has discontinuities (changes in impedance, stubs, branches, discrete elements, other ports, etc.) it is very difficult or impossible to measure the characteristic impedances of multiple transmission line segments within the network.
When people talk about an impedance profile of a trace, they usually mean that they want to see the characteristic impedance of the trace segment along the line. This is impedance vs. position. Z-parameters give use impedance vs. frequency.
In a simple experiment with ports defined at the ends of a single transmission line segment, it is possible to calculate the impedance of the transmission line segment from the S-parameters by plotting on the Smith Chart. If you add one more element then the calculation gets more complicated. The complexity of calculating characteristic impedance grows exponentially with the number of elements in the network. In general, this is not a good method.
A more appropriate, but manual method, is to right-click on each segment in a net and choose the segment properties. This gives you the characteristic impedance of that segment. Record all these characteristic impedances in order and you will get the impedance profile.