Is it dangerous not to connect ground to a 12v power supply
Some appliances may give a shock if there's a wiring fault.   Some, will only give a shock if there's TWO wiring faults.   Each may be
safe if there are no faults.  

But, local safety standards (which differ according to century and
region) may require... almost ANYTHING.

If you might touch the 12V output wiring of the power supply, and if there
is any hazardous AC voltage involved, a ground-fault interrupter
(GFI, or GFCI) and the two-prong plug should make the system
shock-safe.   Any modern three-prong power plug has a ground
that would also make the system shock-safe (if the design passes
modern safety
standards that I'm familiar with).

There is no absolute safety, so there's no possible complete
answer to this question.   Notably, even if there's NO AC power
involved, batteries have been known to catch fire.