Know the Gear Position

Know by passively measuring it"

2 Euro sensor for arduino, measuing the strength of the magnetic field, i.e. derive "gear" from it?

No. Can only measure on or off.

Even if we can measure the different steps, e.g. by measuring the extension of a simple mechanical spring: We can then never automatically change it (which would be cool to later build little apps for recycling scenic routes with hills and stuff)

Know because we actively set it

(mechanically through a motor)

Someone basically rebuilt Shimano's DI2 electronic shifter:

Self Made DI2

We would have to find the best electrical motor for the job (tons of options) and build a "simple" control circuit.

There should be arduino layouts for such a thing.