This is a class II power supply so it does not have direct electrical connection in between DC side and AC side. However, this is a switching supply and there is a capacitive coupling in between primary and secondary windings of the transformer. So they must have included Y2 capacitor in between primary(DC) and secondary(mains). This capacitor conducts current at 50Hz (<mA range) and it is going to fool your DMM on ohms range. Take an AC current meter and measure the current.
As for safety - yes, it is safe to ground class II power supply via scope's probes. But do not try that with class I !! Also, this power supply does not look like a laboratory grade equipment and it might give some unexpected results like excessive ripple, overshoot, serious short circuit currents etc.