It is strange, but probably only if you claim through an insurer, as I did. I don't think it matters whether I had claimed through my own or the other driver's insurance, either way the relevant insurer would have assessed the car for cost and extent of damage, and it then would have been categorised as a Cat B write-off. ie never to return to the road, with DVLA re-registration, MOT and Tax impossible to apply.
If I had not claimed through either insurance company, I assume that they would not have assessed the damage at all. I initially assumed that I could keep the van (considering the damage seemed repairable), so I claimed for the repair bill through my insurer. Unfortunately, and incredibly, they then assessed the damage as a Cat B write off, and I was then never going to put it back on the road.
Legally, they own the car if they pay you for it (as they now have); I could have kept the van but only for spares, and the body would have to have been disposed of at an approved scrapyard - I'm not sure how that works even, as to even remove a Cat B vehicle that you have bought from Copart (the salvage auction company) you have to be licenced to dispose of the Cat B body.
Someone has bought my van, for £80 (plus a million quid of their fees) from Copart today - was it you, Pete Isom?