Can capacitors be damaged by a DMM in continuity mode?
Continuity or Resistance mode applies a very low constant current such as 1mA to 1uA so no such damage will occur at low voltage.  There is not enough power.

However, I suspect leaving a reverse voltage charge on a supercap over time will degrade the capacitor just like leaving a battery undercharged that is not a "deep-state discharge type" will oxidize and damage battery performance over time. (hours)

Therefore if you do test reverse resistance , that's ok for the short term but do not leave a negative voltage on the cap.  So you can expect dV/dt=Ic/C so if 1 Farad and 1mA you expect 1mV per second change in voltage.  Use Voltage mode to check voltage after resistance test shows 0 Ohms

Note: R or diode test mode on a DMM = +ve V on +red
and on  an Analog meter in Resistance mode, the voltage applied is the reverse (+ve V on -ve black) and displayed voltage is shown in R with a log graticule on meter.


  WIKI REF However reverse-charging a supercapacitor lowers its capacity, so it is recommended practice to maintain the polarity resulting from the formation of the electrodes during production.