Magnetism only feels like magic, because we don't have biological sensors for it
Think about it. Isn't light+eyes and ears+sound just the same in terms of their "influence at a distance"? We don't feel that as abnormal or magic - simply because we've sensors for them and are used to it. But physically speaking light and magnetism are based on electromagnetic forces.
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We don't have sensors to directly sense a single magnetic field, but if you've played with magnets a lot, you can definitely "feel" how their forces work and develop kind of an intuitive physical sense.
Perhaps a good example/analogy of something we can feel and understand somewhat intuitively but actually do not scientifically understand is gravity. You can definitely feel its effects / force, even though you don't have a specific gravitational field sensor in your body per say.
We have very good theories and formulas for calculating gravitational effects, but we still fundamentally do not understand what causes it.
I'd argue that the vestibular system in your inner ear qualifies as a gravitational field sensor.
The example with gravity is interesting indeed. We have only acceleration sensors behind our ears, but our body notices the pressure of the body tissue pressing down towards the gravity. And obviously, we also feel gravity when moving.
However, the difference to magnetism is, that we frankly don't have any contact with magnets during our evolution - except for the earth's magnetic field.
Even if we are able to sense it, it's definitely far from being able to reliably feel it like we do for gravity.
Instructions unclear, super glued magnets to my body.