Is all 220 appliance cable the same?
No, not all US 240v receptacles are the same, and none may be right for that tanning bed.  

The color of the wires on the terminal block indicates that this is probably is designed to operate on a European power system that operates one hot leg at ~230v to ground, and the other at 0v to ground. US style power systems operate with two hot opposing ~120v legs, totaling 240v to ground. 

When an 240v appliance (tool, device, or fixture) is designed for European markets the design and safety engineers build their safety considerations into believing only one leg will be hot, and often disconnect switches will only disconnect the single hot leg. Appliances designed for the US market recognize that both legs have voltage potential, and switches will be used that disconnect both legs. So when using an appliance designed for 230v to ground on a US style power system the switch only shuts off half the power.

Why is this important? Let's say one of the bulbs burn out. Somebody turns off the power switch, the lights go out, and they proceed to change a tube. In the process they struggle a little, and a finger, piece of metal trim, or a reflector comes in contact with the lamp socket. If powered by a European 230v system no problem, the switch has turned off all power, but if powered by a US system those terminals at the socket could still be energized, and now the person changing the bulb has been energized. Or let's say heat has damaged a socket, it breaks, now you've got live wires hanging out even though the switch is off.

This is why we look for a Listing mark that is recognized for use in the US. Not all Listing marks are the same. For isntance simple UL with a circle around it and a file number is usually good for the US. Other added marks, letters or symbols (like whatever that symbol is below the "U" on the ballast) indicates tested for some other location or use, like maybe as a replacement component in a product that the finished product has been tested and listed as safe. There are other labs beside UL that are acceptable in the US, the CE mark is not one of them.

There also several configuration of plugs that supply various US 240v power, the configurations will all supply two hots, the first number will designate what additional wires the receptacles has, the second number indicates the amps of the overcurrent protection feeding the receptacle. That number needs to be large enough for the load, but not too high. Let's say you have an 24 lamp tanning bed, 100w per lamp, 2400 watts, at 240 that is 10 amps. You would need to have an outlet designed for at least 10 amps. No problem, the smallest configuration, NEMA 6-15 or NEMA 14-15 is 15 amps. So a dryer plug rated at being able to carry 30 amps should be no problem? What happens if some component becomes damaged and it starts to carry 4x the designed current. 30 amps on a 15 amp the circuit breaker should trip in less than a minute. If you have 40 amps flowing through a 30 amp breaker it could take 10 minutes to trip and still be withing the time curve a UL/NEMA circuit breaker. That could be the difference between smoke and fire.

Then there's the cord. That NM-b cable is absolutely not designed for flexible, exposed, unprotected installations. Cords are rated for size, voltage, temperature, and environmental damage concerns. I'm guessing you need to use a cord that is listed a "sunlight resistant" due to the UV concerns presented by the lamps. I don't know what other concerns may be present, I would absolutely only connect with a cord type recommended by the manufacturer of the tanning bed.