The choice of basic angle in Hipparcos and Gaia is related to the "rigidity" of the stellar reference system that can be constructed when connecting accurate positional measurements across widely separated parts of the sky. Consider the following steps:
(1) The main reason to make measurements from space is to get above the perturbing effects of the Earth's atmosphere (there are also advantages in terms of gravitational flexure of the instrument, thermal stability, and seeing the whole sky - unlike an observatory on the ground).
(2) But a single field of view is not enough to connect accurate measurements in one region of sky (say covering an area of around one square degree) to measurements made in another region. A rigidly connetected reference system across the whole sky is essential for determining absolute star parallaxes and proper motions. Therefore, Hipparcos and Gaia superimpose two fields of view, and make the measurements of two widely-separated fields at the same time.
(3) What should the angular separation of these two fields be? A little thought will show that it should be quite a sizeable fraction of a great circle, say 60-90 degrees. Think of this analogy: imagine trying to create a map of the Earth with a 1-m ruler: local measurements might be tied together extremely well, but the distance from (say) Paris to Warsaw would be hopelessly inaccurate because tiny local errors just build up - it's the same when trying to measure a stellar reference system over the whole sky.
(4) Having agreed that it should be "quite a large angle", are there any constraints on what this angle should be? Yes! Hipparcos and Gaia make their measurements essentially along a long sequence of precessing great circles. Consider measurements made over one great circle (360 degrees) spanning the whole sky. If the two fields were separated by 90 degrees, you would only ever connect FOUR regions over the sky in a single great circle (360/4). If the two fields were separated by 60 degrees, that would be a little better, but you would only ever connect SIX regions of the sky together (360/6). Extending this argument, you can imagine that it is much better to avoid small integer fractions of 360 degrees (1/2, 1/3, 1/4, 1/5, etc.) if we want to create a network of star observations around the great circle with good rigidity. By extension, so too are small integer multiples of these angles, such as 2/3, 2/5, 3/7, etc. of 360 degrees. More formally, angles to be avoided are 360 x m/n, for small integer values of m and n.
Detailed studies in the early 1980s formalised these sorts of arguments, resulting in graphs which show the "rigidity" of the stellar reference system as a function of the basic angle between the two viewing directions. Angles giving poorer great-circle rigidity (90 degrees, 60 degrees, etc) are separated by small 'valley' regions providing good rigidity. Any of these 'valleys' would have worked well for Hipparcos and Gaia, and the choice mainly rested on the physical accommodation of the optical hardware within the payload. Following detailed studies, we chose 58 degrees for Hipparcos, and 106.5 degrees for Gaia. Some more intricate subtleties aside, many other choices in the range 40-140 degrees would have worked equally well.
My weekly essays on Gaia scientific results are available at https://www.michaelperryman.co.uk/gaia-essays, and essay 172 (15 April 2024) on "The Basic Angle" says a little more on this. It includes a graph which illustrates this rigidity versus basic angle more clearly. Reading this answer while inspecting this graph should help your understanding.