Nixie tube and high frequency AC (drawbacks)
The issue is current, not frequency. Nixie tubes require a series impedance to limit current to jut a few milliamperes or they will be destroyed. Glow discharge tubes are damaged by sputtering (atoms knocked out of metal by ions), also called &quot;poisoning&quot;. Doubling the current cuts useful lifespan by roughly a factor of 10!
Humidifier piezoelectric drivers operate at a few watts power, and without a series resistor (or, for AC, capacitor or inductor), the device won't last very long.
That said, there are other issues with using the humidifier power supply:

As others have mentioned, how would you control which elements (digits) are lit? For AC control, you might use an 11-position rotary switch (0-9 and off), but most semiconductor circuits need DC.
N.B. The PCB shown does not appear to be isolated from the AC mains, and would pose a severe electric shock hazard.