There is no guarantee for anything in life and financial stability of an insurer is one of those things, after banking crisis. But it would be taking things out of context if we compare vehicle insurance companies with banks. Banks may have been taking high and unpredictable risks leading to the crisis.
On the other hand, insurance companies are more conservative in the way they invest money. One of the main reasons why they could go down is catastrophic insurance losses. This is unlikely to happen since policies exclude biblical losses. In other words, if something that could have happened in thousand years has happened they could not have predicted it. Therefore, they may can refuse to pay the claims on the basis that it was beyond the scope of insurance.
Although, it may not be reassuring since you may suffer large losses in such cases, it is not hard to understand that this clause is there to protect insurance firms from total wipe off.
Secondly, every state in America regulate sales of automobile insurance policies in their border. So, car insurance companies are regulated by states. They have to apply for a license to offer insurance products. In the same way, brokers and agents need to be licensed as well. States have got to set certain barriers to enter into the market to protect their citizens from rouge companies collecting premium and disappearing. They apply certain capital requirements and tests.
There are certain levels of protections offered by the states. However, you have got to make sure one thing that is the company or the agent you are buying policies from must be licensed in your state.
Also, each state offers insurance safety nets. In other words, they collect money (just like premium) from every insurer who wants to underwrite policies in the state. These funds are used to pay claims if any of the licensed auto insurance companies bankrupts. Generally, all insurers are subject to some sort of regulation and some government body checks their financial soundness in other countries.
How would being insured with one of the top auto insurers help you? Are they safer than the smaller ones due to their size? As mentioned above, there is no too big to fail anymore although governments are more likely to rescue the bigger companies. With the same token, it would be easier to rescue a smaller company and protect the policyholders. So, this question can only be answered when a company actually goes down. If you prefer to work with small and local companies there is no evidence that they are less safer.
People find comfort in name recognition and the number of other people also choosing a particular company. If this is working for you who could say otherwise.