Government mandates only a portion of auto insurance. The only auto insurance that is mandated is liability. You do not have to cover yourself or your stuff. You simply have to cover everyone else against damages you and your vehicle might do to them or their stuff.
So relative to Obama's case, only one portion of auto insurance is mandated by law and it is not the half that has anything to do with protecting the policy owner. (Where is Roe v. Wade privacy when you need it?)
So half of Team Obama's brilliant metaphor caves in right there.
And the other half -- the liability half -- does not apply to health insurance. This analogy would only work if health insurance applied to us injuring or causing others to get sick. Clearly that is not what health insurance does, thus wiping out the other half of the Obama analogy.
This does open up some valid and teachable correlations between auto and health however. They just don't work in favor of Obama Care.