Why (the heck) is the basic angle of GAIA 106.5°?
I did not find anything conclusive on the web, and I am yet waiting to receive feedback from ESA's contact page. In a German speaking IT forum there is something a bit less vague than the FAQ cited in the question:

The reason for the 106.5 degrees is not easy to see. It has to do with the fact that we are (also) doing an astrometric solution on a great circle and for this all angles 360x(n/m) [n, m = small integers] have to be avoided because otherwise one would have a congruence state that would hinder the solution. Values other than 106.5 degrees would also be possible. 90 degrees would be ideal for parallax determination, but unfortunately 90=360x(1/4).

This said, why not simply choosing 91° as $\gcd(360,91)=1$ (with $\gcd$ being the greatest common divisor)?
Maybe it is really only a numerology of the engineers, just like the Mars parachute Morse message. An argument for that would be that Gaia also has also exactly 106 CCD sensors. Or it is simply a logistical issue on how to mount things on the spacecraft. I am really curious what the official ESA statement will be.