Incorrect except for some weird states with 'no fault' and similar regulations which is a function of those states laws and not how proper insurance is supposed to work.
Not sure how long since you were in the industry, but I have been in the collision repair industry since 2001, of those 19-20 years 14 of them were with insurance companies. The deductible is not a separate cost of the claim, it's merely the lowest or first part you as the policy holder are responsible to cover before insurance pays the rest. When the insurance company seeks to recover the costs of a claim they do not request their portion and then say "If you feel like it, our policyholder paid this." If you paid $500 and the insurance company spent $500 they send a demand to the responsible insurance for $1000, not $500 + $500. In fact if the fault for the incident is split and the other insurance is only responsible for 50%, they would only pay your insurance company $500 which was 50%, however your insurance would then pay you back $250, half your deductible. They would not keep the $500 they paid and tell you to pound sand.
Some really good insurance companies will actually give you you're deductible back before they even recover any money from the other company under certain situations. That would not happen with the USPS or any government agency as they operate outside the norm and are not part of the arbitration agreement between many insurance companies.
Anyway, good luck Hungry hopefully the USPS steps up but at least you have good insurance just in case. You said you had Nationwide I think, I worked for them from '01 to '07 and the shop I am at now is a direct repair for them, their parts usage is not what I would prefer but they are a stand up company otherwise.