The "Jazz Bass Mid Scoop" is absolutely real!
When the Jazz bass is played with both pickups at the same volume, the sound will be mid-scooped. The term "mid-scooped" means that the mid frequencies, somewhere between bass and treble, will be reduced in volume.
When the pickups are set to different volumes, the scoop goes away and the mids return. Look at these spectra of notes played on my own 70's Jazz Bass, with both pickups on full volume:
You may notice that the exact location of the mid scoop changes. For this reason, simply boosting the mids on your amp to compensate for the scoop might not work well.
The location of the scoop depends on the pickup placement, string length, and string open pitch, but not on the fret. The Low B string's scoop appears at the same location for every note on the B string, but when you step up to the E string, the scoop's location moves. Look at these spectra of chromatic scales played on one string at a time:
You will notice that the scoop still appears. There are no notes where the scoop doesn't appear that could fill the gap.
What causes this scoop?
This "Jazz bass mid scoop" is a caused by having two pickups at different positions on the string. When a string is plucked, it vibrates at several frequencies, each with different wavelengths. The image below represents the first 7 harmonics of a plucked string.
If one of the harmonics oscillates such that the motion over one pickup is opposite the motion over the other pickup, then the two signals will cancel, and that frequency gets silenced- or "scooped". If you lower the volume on either of your two pickups, then the frequencies will not cancel fully, and you get your mids back.
We can actually calculate the frequency of maximal cancellation. My Jazz bass's open E string is 86.4cm long, so the low E has a wavelength of 172.8cm (twice the string's length). The pickups are placed 10.45cm apart, so the scoop is centered where the wavelength is twice this, or 20.9cm†. This is 20.9/172.8 times the open string's wavelength, so the scoop from cancellation effects occurs where the frequency is 172.8/20.9 times the low E's frequency, or 41.20*(172.8/20.9) ≈ 341 Hz. That matches our image pretty closely! Using this method, we get:
B string scoop - 255 Hz
E string scoop - 341 Hz
A string scoop - 455 Hz
D string scoop - 607 Hz
G string scoop - 810 Hz
You may also notice there are many smaller scoops present too, in the higher frequencies. We expect cancellation at every frequency which has opposite motion over the two pickups, which accounts for these smaller scoops. These smaller scoops are less significant than the first one, but they all contribute to the sound of the Jazz Bass.
You could run these calculations for fretted notes by considering the length and pitch of the fretted string. For example, fret 5 shortens the string to about 64.7cm, and raises the low E's frequency to 55.00Hz (A). Doing the same calculation with these numbers, we get a scoop at 341Hz, which is the same as the open E.
†This uses a hidden assumption that the distance from the bridge to the bridge pickup is half the distance from the bridge pickup to the neck pickup. This is a good assumption for the 70s Jazz Bass, but may not hold for other models, such as the 60s Jazz Bass. For those cases, you can find the scoop qualitatively with a spectrum analyzer, like I did in the images above. The scoop will have similar properties, but it will be located at different frequencies.