In a liability context, the insurer defends the policyholder but pays somebody else (or reimburses for such a payment). (See Lesson 8). But that does not mean that the insurer stands 100% in the shoes of the policyholder.
In many states the claimant does not own any direct right against the insurance company. Schmalfeldt v North Pointe Ins Co, 469 Mich 422 (2003). Though, in other states, there are various rights that a claimant may possess against the liability insurer. Also, after a settlement or judgment, the insured might assign its rights against the insurer to the claimant.
Moreover, the insurance company is not necessarily looking at the entire gamut of the policyholder’s liability. Rather, the insurer is looking to what it is contractually obligated to pay for. If there is liability that is not covered, the insurer might not be willing to just step in and pay out the entire claim.
This is a matter of both the type of liability (not all types of liability will be covered under a policy – Lessons 1, 10), and the amount (policies have limit – Lesson 6).