Two sets of data with different dates in one chart

What chart type are you using? I find that this is very easy on an XY scatter chart, but much more difficult on other chart types.
1) Create an XY scatter chart from one of the data series.
2) Go into the "select data" dialog (chart design group) and add the 2nd data series (both x and y values).
3) Format chart as desired.

The main drawback to using a scatter plot is that a scatter plot does not support the "date" axis that many prefer when the x values are dates. If simplicity overrides this drawback, this is probably the easiest approach.

Is it necessary to be on the same axis system?

One way I can create this combination is to add the secondary axis system. Starting with an XY scatter so that I can readily add the two data series (steps 1 and 2 above)
3) Select series 2 -> format -> plot on secondary axis be sure the secondary x axis also shows up (https://support.office.com/en-us/art...8-e1a2e14dd5a9 I believe that 2010 is similar)
4) change the chart type of series 2 to a line chart
5) change the chart type of series 1 to a column chart
6) Format chart as desired
6a) You will likely want to manually format the secondary axis system so that it matches up with the primary axis system (Axis format dialog -> set min and max values as desired).
6b) If seeing the secondary axis system is bothersome, format the secondary axes so they are less visible or invisible.

It appears to me that you neglected one important step -- the step to add the secondary horizontal category/x axis. In the attachment, all I have done is select chart -> Chart tools -> Axes -> Secondary Horizontal Axis -> Show left to right axis. You will probably want to find this command so you can access it in the future. You will probably also still want to format the axes so that the dates match up closer between the primary and secondary horizontal axis.