The information listed on payable sites such as hpi (other ones are available) is standard across the vehicle leasing, vehicle finance and vehicle rental industry.
The reason it shows the leasing company and details is to prevent fraud, such as where someone tries to sell a vehicle in which they do not have propriety right (ie it is not owned by the seller) or there is still outstanding finance on the vehicle, secured on the vehicle.
Spin it round the other way, if you were a vehicle buyer, you would want to ensure that the vehicle you are purchasing was legitimately being sold by its owner. Else, if you buy it without doing the necessary checks (hpi etc), the actual owner could subsequently come and remove the vehicle from you, as the person/dealer which sold it to you had no right to sell it to you.
You would then be left chasing the seller who could well have disappeared with your money.
So, it is there to both protect buyer and seller and, in this day and age, forms part of the due diligence of buying/selling vehicles.
As a vehicle leasing company, Motability, along with the likes of Lex, Nationwide, Vanarama et al have a duty to protect their assets.
The fact Motability Operations is leasing only to recipients of one of four disability allowances is an unfortunate consequence of this. They are shown as owner (not keeper) as they are the only ones permitted to sell one of their vehicles.