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I know very little about antennas. I know, that for a certain frequency the antenna needs a certain lengths for good sending/receiving conditions.
Often the antenna is at the end of a cable. Why doesn't the length of the cable count in addition to the length of the antenna?

An excellent question! Without diving too deep into the theory, let's start with a few basic terms.

The "signal" that an antenna is receiving or transmitting is called an electromagnetic wave. This is exactly the same type of wave as light. It is just that the our eyes are sensitive to a narrow range of frequencies that we call light. Electromagnetic waves that are lower in frequency behave exactly the same but we cannot see them. These lower frequencies are generally called RF (radio frequencies). Antennas are used to radiate (transmit) and receive electromagnetic RF signals.

The cable or wire that goes between the antenna and a receiver or transmitter is called a transmission line. We use this term even if the cable is simply used for receiving purposes. A transmission line is specifically designed to not radiate an electromagnetic wave but only transport it from one end of the transmission line to the other. There are several type of construction that can meet this requirement but you are probably most familiar with coaxial cable - the same type of cable that is used for "cable TV".

In the case of the coaxial cable, the outer shield keeps the electromagnetic wave contained between the outer shield and the inner conductor. And as it does so, the electromagnetic wave moves from one end of the cable to the other.

The device that generates the electromagnetic wave is called the source. This could be a transmitter or, when receiving a signal, it is the antenna. At the other end of the transmission line is the load. When transmitting, the antenna is considered the load and when receiving it is the receiver that is the load.

So now armed with these few terms, we can restate your question as: Why isn't the transmission line included as part of the antenna when accounting for the length of the antenna?

The simple answer is to realize that an antenna is specifically designed to receive or radiate electromagnetic waves. And you are correct that the dimensions of the antenna play a key role in its overall performance. But now contrast this to the transmission line that is specifically designed to not radiate electromagnetic waves.

We can draw a comparison here to a garden hose and a sprinkler. The garden hose is specifically designed to transport the water from one end to the other without leaking any of it along the way (although I have a few hoses that seem not to have gotten that memo). The sprinkler, on the other hand, is specifically designed to "leak" water in a very specific pattern and volume.

I hope that is the level of answer you were expecting. Feel free to use the comments to ask for any clarifications or additional depth.

Antennas are often resonant. Their physical dimensions are adjusted so standing waves develop at a particular frequency, like a bell rings at a particular tone.

Feedlines are not usually resonant. Usually an engineer ensures the end of the feedline is terminated (by the antenna or the radio) with an impedance that matches the characteristic impedance of the feedline. This ensures that as a wavefront reaches the end of the feedline it's fully absorbed and none of it is reflected back. Since there are ideally no reflections of energy in this configuration, resonance can't happen.

There are of course counterexamples. Non-resonant antennas exist in some applications, especially when the antenna must be physically small, or in receive-only applications where efficiency isn't as important. And transmission lines can be used to introduce controlled phase shifts into the signal, for example to make phased arrays or matching networks, and for this they must be cut to specific lengths that depend on frequency.

In theory the signal should travel down a impedance matched system from the time it has left the transmitter when it leaves the antenna. This is why both the cable and the antenna must be matched or else the signal is defected back from the antenna and wrecks your high priced system.

The Reason why do do not want your cable system to be "Seen" as and extension of your antenna, is that we want to control the radiation that comes from your transmitter in a nice manner.

you are correct that the antenna also serves as a tuning system for your receiver, you can tune your antenna to home in on a frequency.

Overall the antenna is your interface between your radio set and the outside world, the reason why the cable length is not treated as an extension is because, math can be used to optimize the interface between the outside and in the inside. It focuses the radiation an allows for different radiation characteristics that is dependent on the situation.

Imagine this take your garden hose and turn on the water. the water is your signal and the hose is the hose is your cable, notice how the water just fountain and is not "Focused" this is what RF radiation looks like without an antenna.

Now put on a nozzle, this nozzle represents an antenna you can now focus the water in a control manner.

Also uncontrolled emissions of RF signals is bad and would say in violation of your license.

Thanks to everybody answering! This was extremely helpful!