Can you insure someone else's car?

Yes you Can
Yes, You can Insure the property of another Person. So Long as you have authorization to do so, or an insurable interest in that property. Such as a case where dad says " OK, you can use my spare car but you gotta get your own coverage". If You have permissive use from the owner of a vehicle, then obviously you have an insurable interest. However, Only the Legal owner of the Property or the designated agent of the owner can legally receive compensation for the property in the event of a covered loss. If you are insuring for liability only then this is not an issue. If you buy full property coverage, Then Dad will be happy you were prudent enough to fully insure his interest in the vehicle he loaned you.
You can not insure the property of another when no insurable interest exists.

From a Veteran Agent :
Answer
Yes you can insure someone else's car its called "non owners insurance" the problem with that is its only liability.

Answer
It depends on the language in the policy. Some companies allow you to insure a vehicle that is in your care, custody, and control with no requirement of ownership but you must at least be the primary user of that vehicle. I don't know of any case where you would be allowed to insure another persons vehicle in which you do not have care, custody, and control.