I read an autobiography about Vicki Van Meter, who, with her instructor, flew across the United States at age 11 in 1993. Why could someone of that age not drive with the instructor? Does it have to do with a different set of principles that flying has as opposed to driving?
Short answer: the US government agreed with you 3 years after her record flight.
In the FAA reauthorization act of 1996, the following clause was added to 49 U.S. Code § 44724 - Manipulation of flight controls:
(a)Prohibition.—No pilot in command of an aircraft may allow an individual who does not hold—
(1) a valid private pilots certificate issued by the Administrator of the Federal Aviation Administration under part 61 of title 14, Code of Federal Regulations; and (2) the appropriate medical certificate issued by the Administrator under part 67 of such title, to manipulate the controls of an aircraft if the pilot knows or should have known that the individual is attempting to set a record or engage in an aeronautical competition or aeronautical feat, as defined by the Administrator.
Per 14 CFR 91 (can't find the exact text right now, but I know this is true from my flight training), you cannot obtain a student pilot's license until age 14 (so, no soloing until then, let alone a long-distance trip) and a private pilot's license until age 16 (for gliders). You can't get a private pilot's license for your typical single engine aircraft until age 18. A sport pilot's license is possible by age 17, but is limited on the aircraft you can fly (that said, a coast-to-coast trip has been flown in a Piper J-3 Cub, which fits under the sport pilot license) and conditions you can fly in.
Hence, it would not presently be legal to do what Vicki did in 1993. According to a NY Times article covering another young pilot, Jessica Dubroff (aged 7 at the time of her fatally unsuccessful attempt at a coast-to-coast flight):
Jessica was breaking no aviation rules when she flew. The plane had dual sets of controls, meaning that the flight instructor was legally considered in charge and had the ability to take control instantly, though the plans called for Mr. Reid to take over only in an emergency. But within hours of the crash, Federal authorities announced that they would re-examine existing regulations on underage pilots.
This would have preceded the addition to the USC quoted above.
I can't give an authoritative answer because I doubt one exists, but one huge difference between cars and planes is that all aircraft used for primary instruction can be flown from either seat. Both legally, and physically.
It's almost always entirely possible for a pilot to conduct an entire flight by themselves from the second seat, save perhaps some small logistical issues with the location of a switch or two.
To that end, it's not possible to fly in the front seat of a trainer without being at the controls. Therefore any passenger, in either seat, is sat in a position of being able to take control of the aircraft.
In addition to that, while aircraft are often designed with one primary seat in mind (normally the left hand seat), there are rarely any legal restrictions that say a training aircraft can't be commanded from the right.
This differs hugely from a car which, firstly, is designed specifically to be driven from one side or another. An instructor in the passenger seat may have a SUBSET of controls, but sometimes not and therefore can't fully control the vehicle from that seat.
Driving requires a much faster reaction time. Look at how the consensus process works inside a cockpit under healthy CRM. Just having a brief "My airplane" / "Your airplane" is far too slow for typical driving events.
Even if we set aside those formalities and simply have the Co-driver "leap in when needed", events don't unfold that way. Actually what happens is the Co-driver sees the emerging problem (truck pulling into their path) and the first thing she does is presume the Driver is going to handle it correctly. If the Co-driver leaps in too soon, 99% of the time she ends up needlessly wrestling the driver for control. So the co-driver intervenes quite late, only when it's clear the Driver isn't doing the job. At this point, usually, your goose is cooked.
A classic example of this was the first fatal crash of someone letting the computer drive: the co-driver was attentive and not watching a Harry Potter movie. Things unfold in rail operations slower than they do in cars, and yet still, discovering the driving mistake happened too late to prevent it.
All this to say... events in an airplane typically happen a lot slower, giving pilots a lot more time to use CRM to work solutions. Once the solution is agreed, the hard part - the use of judgment is done, and it's down to execution.
Regarding the comparison of the "set of principles that flying has as opposed to driving":
It's not necessarily the difference in the principles that matters; both kinds of vehicles have ways to control how fast you go and the direction in which you go, and in both vehicles it is possible to lose control (and sometimes also to regain control). But the principles of flight and the principles of driving are typically exercised under vastly different sets of conditions.
If pilots routinely flew along routes where there were solid obstacles just a few meters from the wingtips on either side (like the trees lining a country road) or if there were typically other aircraft flying in the same direction three seconds ahead of you and three seconds behind, and sometimes someone passing you in the same (or opposite) direction just a few meters away, I imagine the rules of flight would be somewhat different than they are now. But those are not the usual conditions under which people fly.
A very large part of the population drives, which means that a large part of the population consider themselves authorities on driving, and qualified to tell other drivers what to do. Far fewer people are pilots, which means that the average person more or less trusts the professionals to know what they're doing when it comes to flying. This results in different regulatory environments.