Is this correct or the current rating would be close to the maximum
pulsed current limit of 160 A during the on-period (<10 us)
It is correct that the device can withstand 1,000 volts at about 23 amps for no more than 10 μs but, you'd be wise not to get within about 50% of the implied instantaneous power of 25 kW.
As for the 160 amps limit, that entirely depends on what your circuit produces.
So, when the MOSFET is activated (i.e. fully "on" as defined in the data sheet) it will withstand 160 amps but it certainly won't withstand 160 amps as the device is in the process of switching from off to fully on or vice versa.
My confusion stems from the fact that even at 1000 V supply voltage,
the voltage across the FET should be pretty low during on period
(Vgs=20 V).
A 1,000 volt supply voltage isn't the same as \$V_{DS}\$. The SOA graph has an x-axis of \$V_{DS}\$.
Here's a bit of advice; model your MOSFET and simulate the circuit and look very closely at the peak values of \$V_{DS}\$ and \$I_D\$. In fact, I would highly recommend that you use your simulator to calculate \$V_{DS}\times I_D\$ and look at that plot directly.
And finally, you should be aware that these devices will go "pop" during the transition from on to off and vice versa if you do not engineer a decent gate drive circuit. They are capable of switching in 20 ns or less and you should not entertain anything much slower than a 100 ns transition time if operating at 1,000 volts with tens of amps.
They are a good device BTW and good luck in getting hold of them. I placed an order in April this year with mouser and, at the time, they reported a delivery of late July this year but none arrived and the delivery date has fallen into mid 2022. I chose a different device!!
Don't forget to place a cut-out in the PCB around the drain so that 1,000 volts can be reliably switched.