Hypothetically, why can't we wrap copper wires around car axles and turn them into electromagnets to help charge the batteries? We already have a magnetic core, why can't we use it to recharge the batteries? The only problems I see with it are potentially wiping magnetic data, but doesn't the electromagnet have to be revolving around the damageable device?
 A: This is basically what happens in the alternator. The car's engine, which turns the wheels, also turns the alternator's rotor. The magnetic rotor is surrounded by coils of wire, and induces a current that charges the battery. It is important to note, though, that this does take energy from the engine: nothing is for free.
A: Yes. This is how regenerative braking works.
There is a style of brakes that is used in many electric or hybrid cars that, rather than converting the motion of the wheels into heat energy the way conventional brakes do, instead convert the motion of the wheels into electrical energy that recharges the car's battery. By doing this, they can improve the performance of the car and extend how far it is capable of traveling when driven in environments with frequent braking, like most cities.
There's no technical reason why such a braking system couldn't be installed on a car with a purely internal combustion engine, though I expect that the energy that they produce might be wasted once the battery has been fully recharged, and the extra cost probably isn't worth the marginally increased battery performance.
A: Not stated in your question is the type of vehicle we are discussing - I am assuming this is about vehicles powered by internal combustion engines.
Attaching some sort of electricity generating device to the car's final drive to recover kinetic energy during braking comes down to cost/benefit.
Every piece of equipment you add to a car increases its mass, which reduces its overall efficiency - more fuel consumed to haul it all around.
Car engines are equipped with alternators to provide electrical power whenever the engine is running, in part because the car doesn't have to be moving in order to power lights and accessories, charge the battery for the next start, and so forth.
The amount of power moving around the electrical system and the amount of energy stored in an automotive battery is relatively small compared to the kinetic energy of a moving vehicle and power required to get one up to highway speed in an acceptable amount of time.
An energy recovery system would need to be rather sophisticated to capture a useful amount of energy from a decelerating car - the slower it is moving, the less efficiently it will work.
Electric cars use regenerative braking, it works, and is worthwhile because the motors used to drive the vehicle are operated in reverse; the same strategies which allow the motors to efficiently accelerate the vehicle over a wide range of speeds allow for efficient energy recovery when decelerating. It's a matter of re-using equipment that is already there.
In summary, the existing alternator meets the need and is well-matched; attempting to recover kinetic energy during braking as electrical energy would be too poorly matched to the requirements of the electrical system to be of practical benefit for the cost incurred.
A: The car's alternator would be more efficient and economical way to charge the battery. If you had generators or alternators on the wheels the engine would have to work harder to turn the wheels as there is no way to receive more energy from a machine than is put into it.
A: This is a classic "Free Energy" setup, which sounds great...until you realise that crummy ol' conservation of energy.
The problem is the induction principle. It's what makes electromagnetic motors possible, but it's what stops us from free energy. The problems is that the faster the wheel spins, the harder the magnet is going to resist that spin as it generates electric current.
Because the magnet is going to resist the spinning of the axle, which would be what makes wheels rotate, you would always need to have more power from your engine than the power which the electromagnetic axle generates. In essence you would generate some electricity in your axle, sure...but you would always have to use more power from your engine.
You can run through the math and whatnot, but that is the basic idea.No such thing as a free lunch, especially when it comes to energy!
A: Because of how electromagnetism works, plus a lot of practical issues
Other answers have focused on what is currently available and why that works. I'm going to aim at the deficiencies of your exact idea.
The first problem is that simply wrapping wire around the axle won't achieve anything, because in order to generate electricity, the coil needs to be perpendicular to the direction of motion. So we're looking at coils laid along the axle somehow. Not only is this going to be difficult to achieve, but it'll create all sorts of problems with driveshaft balancing.
Then we have the problem of how a generator works. You don't just need a coil, you also need a separate magnet as well. This could be a fixed magnet, or an electromagnet. One part of this is fixed (the "stator") and the other (the "rotor") is attached to the spinning axle. You can choose whether you want the generating coil to be the rotor or stator, or the magnet, depending on how you want to build this. So we need something big and magnetic attached to the axle - which in a front-wheel-drive car is part of the unsprung weight of the car, by the way, so that's undesirable. Let's skip that detail and say we've got a rear-wheel drive car with a driveshaft though.
Then you hit practical problems. Axles and driveshafts are in an area of the car which is very prone to flying stones and other debris. Not a good place for delicate electrics. They also twist substantially under load, which is not good news for anything fastened to them.
But the real killer is electrical. The efficiency of magnetic coupling depends on how close together the magnet and coil are. Driveshafts will inevitably wobble under load, so you'd need the stator to be spaced away to allow room for that. This is a major blow to your efficiency.
To make matters worse, electrical requirements mean the coil needs to be a lot of turns of very fine, very thin wire. I mentioned stone strikes earlier. These are going to be "very* vulnerable. Sure, you could put a big casing around it for protection - and this is all starting to get not very practical.
Compare and contrast to an alternator. Because it doesn't have the same mechanical constraints, it can be built to put the rotor and stator close together, with fine wire, all inside a conveniently sized package. That's why we do it that way.
And finally, it'd only work when the car is moving. Unless you have an alternator connected to the engine as well, you're going to run out of battery if you're stuck in traffic. And if you've already got an alternator, this is redundant.
A: 
We already have a magnetic core, why can't we use it to recharge the batteries? The only problems I see with it are potentially wiping magnetic data, but doesn't the electromagnet have to be revolving around the damageable device?

This is precisely how cars both charge their batteries and produce electricity to run things like lights and the radio. The magnetic core can be anywhere in the power train between the engine and the wheels. Generally, it's easiest to connect it to the belt that already runs between the engine, the air conditioning compressor, the power steering pump and sometimes other pumps.
The device that does this is called an "alternator".
More modern cars also use this to convert excess energy to electrical energy during deceleration or going downhill.
