It might be a good idea to start by trying to make a block diagram of what you have inside of each of those controllers...
Namely, the drone controller is most likely some sort of a system on a chip, which on its own chip, in addition to having a microcontroller also has the radio interface built in.
Those are two separate things, but in one package.
Radio modules usually only communicate over radio, and need a extra digital electronics to drive them (today commonly implemented using microcontroller), which lead to the popularity of specialized microcontrollers with integrated radio interfaces. There are also many other types of specialized microcontrollers, with Bluetooth radios integrated, with audio amplifiers integrated, with FM and DAB radio receivers integrated and so on.
On the other hand, Arduino is a platform for rapid prototyping, which is based on Atmel's (now Microchip's) ATmega series general purpose microcontrollers. Its main selling point is not that it's especially cheap, or that it has integrated modules for something, but that it's super simple and relatively easy to understand, and therefore attractive to beginners.
As I mentioned, the ATmegas are general purpose micro controllers. They do not include the radio interface in them. Therefore, you need to use an external radio interface. That interface, depending on its complexity, will most likely converts the signal into data, which the microcontroller can easily process. However, since the interface chip doesn't know what is it actually going to be talking to, it needs to implement a lot of intelligence on its own, in order to covers the radio signal to data, which drives the cost of such modules up.
Now, let's go to the "piece of wire" statement. The piece of wire is almost certainly not just some piece of wire, but a wire antenna, hopefully cut to the appropriate length. The drone controller's PCB also most likely has tuning elements on it, which should allow it to properly drive the wire antenna you have.
The radio interface module for the microcontroller will also have its own antenna, either on the PCB, or perhaps externally, and needed electronics to drive it.
And finally, one important thing to keep in mind: In order to have any hope of capturing the drone's signal, you need to actually figure out which protocol the drone uses in its controller, and to know which modules can work with them, so that you can select an appropriate one for your microcontroller. Just picking one up at random is not going to work, since the protocols running at 2.4 GHz tend to be very complicated.