I expect it is finding a way to measure multiple frequencies around the tuned station and decodes the data from it’s deviation from the tuned frequency?

Yep, and that’s bandwidth. Up to 200kHz for WFM I think.

Edit: I’ll add some images. Maybe they could be helpful.

This is screenshot of RF spectrum. Frequency increases to the right, time towards bottom. On the left is some regular FM station, on the right is signal from my cheap FM transmitter. It’s mono-only making it perfect for this test.
The white rectangles show filter bandwidth and red line center frequency. That’s what you tune, center frequency.

In this image it’s just transmitting unmodulated carrier wave. In other words, silence.

Here I modulated it with a simple sine wave. I slowly increased its frequency over time. Probably seems familiar. (It looks like dots because the spectrogram is drawn at 20FPS. With high enough framerate it would be a continuous line.) Frequency dictates how often the carrier wave swings.

Here we’re back to 1Hz sine wave, but instead of changing the frequency, I changed its amplitude (volume) up. The further the frequency deviates, the higher the amplitude. (Again, 20FPS. It’s not changing the frequency suddenly like that.)

For the sake of making it a continuous line as it is, here it’s shown at 600FPS.

You can see the FM station at the left starts to fall apart into waveform a bit as well.

Of course, music, speech or whatever isn’t a single sine wave, it’s a sum of range of frequencies of different amplitudes so it doesn’t look that nice.

Of course, don’t trust me on anything. I likely said something wrong.

The FM tuner is a narrow band pass filter centered at the carrier frequency - it doesn’t try to track the modulation. The FM demodulator then uses a VCO (voltage controlled oscillator) with a phase detector in a feedback loop, so that the output of the VCO tracks the modulated input signal. The input to the VCO then represents the baseband (demodulated) signal.

RF only has 2 components, Phase (frequency) and its amplitude. For Analog FM radio, you have a center frequency you tune to. The variance from the center frequency (phase) is the amplitude of the carried signal. For digital signals, you will have specified offsets from the center that represent specific binary codes.

Edit: as others have said, the tuning and demodulating are 2 different steps. Step 1 tune, When you tune you take the signal centered at the carrier, what the dial on your radio says, and recenter it at 0. Step 1 is the same for pretty much everything RF. The output is “base-band”. You aren’t going in and out of tune because for each center frequency, there will be an agreed variance (band width) allowed for the channel. The tuner captures this entire range and this is what is then demodulated in step 2.

This isn’t quite an ELI5, but ARRL has a 2004 article on FM fundamentals; it’s five pages intended for a beginner ham radio operator, but applicable to all FM applications nevertheless. It also discusses four different ways to receive FM.

But to answer your question directly:

The frequency of the FM signal at any instant in time is called the instantaneous frequency. The variations back and forth around the carrier frequency are known as deviation

FM can also be detected by a PLL. As shown in Figure 6, the PLL’s natural function of tracking a changing input frequency can be employed to generate a voltage that varies as the input frequency change

In a nutshell, FM only ever has one instantaneous frequency at a time, which dances around the nominal center frequency (aka carrier). So the receiver has to detect the instantaneous frequency, relative to the carrier.

To actually recover the original signal, the receiver must also account for the modulation index used by the transmitter, which describes how much the output will deviate for a given input frequency. The modulation index is usually standardized for the application, such as FM broadcasting, amateur radio FM, walkie talkie FM, etc.

Because a larger modulation index means the same input signal will result in wider deviations, more RF bandwidth is used, spreading the signal wider and generally improving noise immunity.