I have designed a Sallen-Key bandpass filter with a centre frequency of 5000 Hz, a Q factor of 2 and an H0 of 5.
A square wave of 200 Hz contains odd harmonics all the way to infinity. The 25th harmonic of 200 Hz is 5 kHz hence you see the band pass filter extracting this harmonic and greatly amplifying it.
Here's a picture of the spectrum of a band pass filter on the 17th harmonic (nearest I could find): -
Picture taken from Acoustics and Psychoacoustics: Introduction to sound - Part 8
So, if you look at the step response of a band pass filter you will see waveforms like this: -
Your circuit (Q = 2) will have a zeta of 0.25 (pretty close the the green trace) and each time the square wave pattern changes you get the step response repeating again and again.
Your interpretation is flawed - a sallen key filter isn't the same as two cascaded 1st order filters because two cascaded filters cannot, by mathematical or practical definition, produce a Q greater than 0.5 and therefore cannot generate decaying sinewave responses as you see.