What is happening physiologically to cause this shaking?
Sorry, I can only speculate on this.
Is the intensity of the shaking cumulative, such that it will reduce over time with regular practice? Or does it just depend on how tense you are at the particular time you do it?
The shaking goes away with repeated practice sessions. It goes away because you learn not to fight yourself, not because your muscles get stronger, similarly to how a ball player with metaphorical stone hands learns to catch better by relaxing.
The most basic qi gong posture is simply standing with your knees slightly bent and your arms down by your sides. In a practice session you stand still in one place for long enough that your muscles somewhere get tired. This is your indication of where your body is not aligned well with gravity and you are using muscular tension to make. You adjust your posture both in the moment and across sessions to try to reduce the muscular tension needed to maintain the standing position. Over sustained practice, your posture improves and the act of standing becomes easier. In some sense, you are simply learning to relax or rest better, and practicing basic posture should become pleasant.
In my experience, the "qi blockages" are not so much about shaking, as the difference in body feeling between the state where you are holding the body up with excess muscular tension and not. There is a spectrum of such conditions, but when you first feel the muscles relax, there can be a feeling like you have had a great massage or sauna session.
Not every posture should be used for standing qigong. The horse stance, for example, can be done in multiple ways. In wushu, the ideal is with the thighs parallel to the floor. This develops a great deal of leg strength, but it would be a waste of time to use this for a standing qigong session because it is impossible to hold this position without a great deal of muscular tension. If, however, you create an arch with your legs, your structure carries your weight and the muscles just have to hold the bones in place. In this version of the stance, you can learn to use less muscular strength.
The standard qigong advice is that you should not try to intentionally recreate events like this, but simply experience them.