These are just my thoughts. I am sure there are others who routinely use calibration kits, who may fix what I get wrong..
λ/8 and 3λ/8, if doubled, give odd-numbered multiples of a quarter-wavelength. Doubled means the round trip to the open or shorted end, a 100% reflection, and then back.
A quarter-wavelength guide, if open circuit, will present a short-circuit at the other end, and vice-versa.
The 100% reflection arrives back 180 degrees out of phase, and cancels the wave it found at the connect port.
A λ/8 or 3λ/8 will do something else. A 3λ/8 length, if other end open, involves a round trip of 6λ/8 = 3λ/4.
A wave arriving at an open circuit finds itself on the end of a transmission line with no place to go except back from whence it came, down a transmission line line with boundary conditions and impedance that make propagation in that direction possible. This is true of both waveguide and coaxial sorts. Only the modes are different.
A wave arriving at a short-circuit is also 100% reflected, except that it comes back "upside down". The physically impossible requirement to run a infinite current into the short results in a field voltage that exactly cancels the arriving field, and returns as if it was 180 degrees out of phase at the short. This is the exact opposite of the arrival at an open circuit, where the impossible condition is to have zero current, or infinite voltage. This property is the basis of all TDR (Time-Domain-Reflectometer) measurements.
The reflection is not perfect. In the case of coaxial, it very nearly is. In the case of waveguide, some is transmitted out the end as radiation field into a 377 Ohm mismatch. It occurs to me that a good calibration open circuit might be contrived out of short-circuited parts of special lengths. It just might be a procedure where first using a λ/8, and then a 3λ/8 built for one frequency, and measuring the phases for the frequency sweep, might provide the information to the analyser to then be able to calibrate for all other frequencies in the sweep. I speculate here. Its the only thing I can think of.
I know that really good open and short circuits are expensive, and probably quite difficult to make. I accept that the above does not directly answer your questions.
Now both of us can wait and see if there are members who can provide a clear explanation of why the λ/8 and 3λ/8 are so specially useful, and how to use them for calibration..