Is it possible for a plane to be over fueled (and over-weight) to take off?
The aircraft you flew on was D-AIUX, a Lufthansa Airbus A320-214. On 10 February it was scheduled to fly from Munich to London, but the flight was cancelled due to the storm:


(flightradar24.com)

If the aircraft was fueled for this flight the day before, it would need:


3.13 kg/km x 944 km = 2.95 t of fuel for the trip
about 1.5 t of reserve fuel


After actually flying the 356 km to Vienna, it would have burned about 1.11 t of fuel, resulting in a total weight of:


42.6 t OEW (Operating Empty Weight)
3.34 t of fuel remaining
+any payload


The MTOW (Maximum TakeOff Weight) is 78 t, which would therefore still leave 32.06 t for payload, but the maximum payload is 19.9 t for the A320. So this cannot be the limiting factor.

It was noted that the runway could limit the takeoff. Your flight departed from runway 29, which is 3500 m long:


(flightradar24.com)

The A320 only needs 2100 m at MTOW for takeoff, so this was also not the limiting factor.

You said "they filled the plane in Munich the day before to give the plane more balast to cope with the storm winds", which is not a procedure I am familiar with. Increasing the aircraft weight will increase its inertia and therefore make it less susceptible to wind gusts. Let us assume they completely filled all fuel tanks in Munich.

The A320 has a fuel capacity of up to 27,200 l. Multiplying that with the density of Jet A-1 (0.804 kg/l at 15°C), gives about 21.87 t. This would make the total weight before payload at Vienna 63.36 t, therefore leaving 14.64 t for payload. Now this is lower than the maximum of 19.9 t. Using standard passenger weights in Winter of 195 lbs = 88.5 kg, this would allow 165 passengers. The A320 exit limit is 195 passengers. It is therefore possible that some passengers had to leave the plane to get below MTOW.