I plan to write a devlog post about this at some point, but at one time I wondered the same thing. It turns out that starting in 2D, when there is more than 1 spring, things get more complex than you expect: it turns non-linear. If the mass is moved exactly along the axis of the 2 springs in the demo you linked, then it's just a simple harmonic oscillator. But if it's moved OFF axis, the force from the springs is constantly changing direction. The force orthogonal to the spring's resting axis increases with the sine of the angle that the spring has been pulled off to. So along the off-axis it is not a harmonic oscillator.

If this doesn't make sense it's entirely my fault for explaining it poorly -- I really want to write this up properly with some diagrams because it's a surprising thing and quite interesting, at least to me.

(BTW that demo has other things going on like mic isotropy and analog oscillators attached to vibrate two of the masses, so my spring tangent is not the only thing producing harmonics.)