Is that somehow technically possible?

Is it possible to split the sound of his laptop, so that you hear because of You Tube on headphones and at the same time a radio station on the laptop speakers?

Theoretically already with USB headphones. Because these count as independent audio output.

Normal jack headphones push themselves in front of the right speakers, so it does not work!

Then you only need a program where you can choose the audio output.
For example, if you've set the speakers as the default and are watching Youtube through the browser, you're hearing about the speakers.

If you have a stream file from the radio, you can play it in the VLC player and select the headphones as the source.

I hear with a USB headset, but what kind of program is that supposed to be?

As I said, you can select the speakers from the laptop as standard.
Your internet browser will always use the standard.

In the VLC player you can then select as the audio output the headphones and play a streaming file.

Technically it is possible, practically not. You could use such a sound card but in most cases a simple soundcard is installed, which can only output one stereo signal. There are cases, such as DJ programs such as Virtual DJ, because you can send two different mono signals. One for the speakers and one for the headphones to listen to.
So you see, it is theoretically possible but the software has to be able to do that and it would only be mono.

Or you need programs where you can choose the audio output.
This will directly address another sound card.

Fortunately, it has USB headphones, so they are listed as another device.

Here is just an example. For me, unfortunately, now only "Remoteaudio" but you should be listed with the headphones.

Simply select these in the VLC player and this sound is about it.

Do not you need a second sound card for virtual DJ? Once had a program where you need a second sound card.

It's a second soundcard / dac pretty easy. Are yes USB sound cards already for a few euro.

Bluetooth or USB headsets have their own dac, so it works.

Then you only have to choose the output in the program.

Virtual DJ can put on the 2 channels each sound card, namely left and right, different signals, which are then of course Mono. So just for example Once master and once headphones. With a suitable adapter from stereo jack to 2 times mono jack you can use headphones for preview.