What climb rates can the Airbus A320-200 achieve and which climb rates are commonly used for normal flight operations?
There's a lot of nonsense being banded about regarding this flight.
First up, we are not talking about sustained, powered climb rates. If you're travelling at Mach 0.85 and pull up vertically, in theory (if your plane didn't break) you'd climb at an initial rate of nearly Mach 0.85 (your original speed minus the airspeed loss when pulling up).... That's near enough 57,000 feet/minute.
If you're at high speed and willing to trade speed for altitude, an Airbus A320 is physically capable of climbing at much higher rates than 6000 feet per minute.... At least until it runs out of airspeed.
A fighter is the only aircraft which would be likely to perform a sustained 6000ft/minute climb rate in normal use, but they're also capable of much higher rates of climb.
The maximum rate of climb you'd see in an A320 in normal use is in the 2000 ft/min range, but that's because it's the highest reasonably efficient rate of climb. That's not the maximum the aircraft is capable of under power (which is more in the region of 4000 ft/min), and is absolutely no indication of that maximum unsustained climb rate.
It would eventually stall at any unsustainable rate of climb, but then again, so would any aircraft even at a sustainable rate... Keep climbing and before too long you run out of atmosphere.
Either way, there is nothing to say that climbing above the sustained climb rate for an aircraft is unsafe or, necessarily, unusual. You trade airspeed for altitude, sure, but if you need to be higher at that moment in time, there's plenty of time to accelerate again later.
From Airlines.Net:

The three are roughly the same at 10,000', at about 2500 fpm. (Maybe a bit more for a light A319, and less for a full A321)
Normal climb rates from 10,000' to about 30,000 are as follows:
A319, 3000 fpm to about 1500 at 30,000'
A320, 2200 fpm to about 1000 at 30,000'
A321, 1400 fpm to about 500 at 30,000'

TL;DR - The official is spouting nonsense s/he doesn't understand. A fighter is the only aircraft likely to be able to (or wish to) climb at a 6000 ft/minute SUSTAINED climb rate, but almost any aircraft can climb at that rate for brief durations in a zoom climb - converting airspeed/momentum to altitude