What does the truing "release tension" step do?
This subject can be confusing to learn about because people use terminology to mean different things, and some of the physical techniques involved are working to help achieve a couple different goals simultaneously.

When truing wheels, and especially when doing extensive wheel truing or building a wheel, you must be concerned with managing spoke windup, and your goal is always to have as little as possible when you're finished. Good mechanics get it to effectively zero, but it's a skill that takes practice and attentiveness, and there are a couple ways to go about achieving it depending on the situation. I believe the tension you're referring to releasing is someone's confusing choice of terminology for relieving spoke windup.

Spoke windup is when the spoke gets twisted as the nipple is turned and gets held in place that way by the spoke tension. Then when the wheel rolls along and the spoke tension gets unloaded, they'll pop back and the wheel can go slightly out of true (or more than slightly if there's a lot of windup). The release of that torsional load is presumably what you're referring to. You want to get it all gone before the wheel gets put back on the bike.

In actual practice, controlling windup is mostly about very carefully overshooting and backturning your nipple adjustments. With most spokes (2.0, 1.8, and 1.7) it's usually sufficient to only do this the last time you touch each spoke, in my experience. For 1.6 and less I do it every step of the way. (For bladed spokes you get to use a tool to keep them from twisting, so windup isn't an issue.)

Sideloading the wheel by putting it on the ground and pushing on the rim as you describe relieves windup by temporarily unloading groups of spokes, allowing you to recheck it. It's also a good thing to do to new machine-built wheels to keep them from going out of true immediately. It's also a way to destroy wheels if you're not careful, or if you've got too much tension on them. Some people claim that sideloading can have an effect on setting spoke line at the hub to help prevent spoke breakage, but I don't really believe it, and to do that part right at any rate the spoke line has to be set when the spokes are at no or low tension.

It's important to understand that sideloading is more like a quality control step than your primary tool to get windup out, which is your technique with the spoke wrench.

Gripping pairs of spokes and squeezing them has more to do with stress relieving spokes to prevent breakage and also helping get them mated with the hub in their final position, which helps prevent the wheel from becoming untrue.