When I try to "double-click the plot area to open the PTC Mathcad Chart application" nothing happens.
Keep in mind that the chart component is a badly integrated third party tool which is quite crappy and incredible slow. So when you double click it can take quite a long time until you see any reaction. Furthermore the popup window which open upon the double click ever so often opens beneath the main Prime window, so you may not notice it until you are actively looking for it.
Of course you know that you cannot double click a native Prime 2D-plot and expect the chart component to po up, don't you?
(i.e. no chart application shows up). In the Axes section of the menu bar there isn't anything that addresses adding secondary axes.
Not sure which "Axes section" you are talking about? But reading this now I fear that you really are not aware that the native Prime plots you create with "Plots"-"Insert Plot"-"XY Plot" have nothing to do with the chart component and its also not possible to convert a plot created that way into a chart plot.
To insert a chart plot you will have to use "Math"-"Chart Component"-"Insert Chart Component". The plots created that way can't use units, don't scale if you have a 4K display and are very awkward and cumbersome to edit. But they have the benefit of a secondary y-axis, gridlines, etc. which PTC wasn't able or willing to implement in Primes native 2D plots. Ridiculous, given that PTC sells a full-blown CAD software, but unfortunately true fact.
If you really have to stick with Prime and need a secondary y-axis, you'll have to give the chart component a try.
P.S.: You may want to study the section about the Chart component in the Prime help: