Can we able to measure the gain of any test antenna on VNA with out using standard gain measurement set up procedure (two antennas: one is AUT and another one is horn antenna).
Yes, if you have two similar test antennas just apply Friis formula to them in a lab link- the gain will be squared since they're the same, all other terms will be known, you'll get their gains.
Pr=(Pt.G^2)/(4.pi.d/lambda)^2 This is friis. G is the gain of the antennas (two similar ones, with the same gain). You place them at a known distance d (should be farfield), excite one of them with a power source of known frequency (lambda) and power (the Pt) and measure on the other end with a spectrum analyzer the received power.
You can take a look into Balanis Book, Antenna Theory, on the equation, chapter 2.