You may be thinking of insurance that is paid as part of your mortgage payment. Mortgage companies may set up an escrow account that you pay as part of your mortgage payment and they forward the insurance money to the insurance company. It is still your insurance policy even though the mortgage company sends the check.
Sometimes when homeowners let thier policies lapse the mortgage company will get a policy that is called "forced place insurance". Those polices are obtained by the mortgage company to protect thier interest in the house but you still pay for it. They will increase your mortgage payments to offset the cost. Remember, forced place insurance only protects the banks interest even if you are the one paying for it. It is always better and almost always less expensive to shop around and purchase your own policy. You get way better coverage for a lot less.
If you stop making payments, neither the mortgage company or servicer gets paid. The exception in most cases is the lender will continue to pay the property taxes and hazard insurance on the property if you fail to keep coverage.... more