From what I have understood from a co-worker that had a collectible 'muscle' car damaged/totaled, the insurance company settles with you for an agreed upon price, they issue you a check then the insurance company 'owns' the vehicle, then you purchase the vehicle back from the insurance company .
Usually you end up with a 'salvage title' for the vehicle, that can make it a bit difficult to sell, if you ever had or needed to .