I've made some progress in vector mode using EddyCurrent's suggestions.
What I did:
Set PB.02 -> 2.2 (Motor rated power)
Set PB.03 -> 100 instead of 400 (Motor rated frequency)
Set PB.04 -> 6000 instead of 24000 (Motor rated speed(RPM))
Set PB.05 -> 55 instead of 220 (Motor rated voltage)
Set PB.06 ->10 (Motor rated current)

Then I run autotuning (PB.00 -> 1)
After autotuning finishes:
Set PB.08 -> 00.050 (Motor rotor resistance). This is the most important! . After autotuning it is set to 00.000.
Then, without touching PB.02 (power rating. if changed, it resets all detected tuning parameters):
Set PB.03 -> 400 (Motor rated frequency)
Set PB.04 -> 24000 (Motor rated speed(RPM))
Set PB.05 -> 220 (Motor rated voltage)

After that it seems to work in vector mode. A few notes though:
1. It won't spin high than around 14000rpm (~233Hz). When approaching this speed or going beyond, it oscillate for a few seconds around 14000rpm and then steadies itself at this speed.
1. I can feel it increases torque at low rpm. When in v/f mode at ~1000rpm I can easily stop the spindle with my hand (in a glove), when in vector mode it feels like I have to use 2-3 times more force to stop the spindle. Is this how much it should be expected from vector mode?