Can chicken soup be frozen twice?

The short answer is yes, if you bring it to a boil (and cool down) before refreezing.

Appreciate the explanation. Thanks!

Think about freezing as putting an item in suspended animation (because, with only very minor exceptions, that's exactly what freezing does). Bacteria grow slowly < 40F, not at all < 32F so safety is determined by total accumulated unfrozen time.

The general rule for leftovers is 4 days under refrigeration. It doesn't matter if that's 4 consecutive days or 1 day thawed, a week in the freezer followed by 1 day thawed, etc. The goal is to prevent a buildup of bacteria to the point where they could become an issue.

It's always a good idea to reheat leftovers, frozen or not, to 165F before consumption but it's not a necessary step, just a secondary precaution. For example, eating a cold 4-day-old piece of fried chicken or slice of pizza is perfectly safe. Reheating -- or, technically, re-pasteurizing -- has another advantage in that it resets the 4-day counter by destroying any active bacteria present. In theory, given proper times and temperatures, one could keep a freeze-thaw-reheat-freeze cycle going indefinitely.

Food quality is a different issue although a much more minor one than most of us have been led to believe. Freezing causes damage on a cellular level and repeated freezing will cause increasing damage due to moisture loss. But the damage is minor, on the order of 5% or so, which is all but undetectable in most foods. In this example, chicken soup, so much damage has already been done to the meat and vegetables by cooking that a few freezings shouldn't be a cause for concern.

Chef Uno, wouldn't the safety issue depend on the product being thoroughly reheated? I have always avoided multiple freeze / thaw cycles for two reasons. First because of the safety issue, but second because reprocessing can cause product degradation.

Multiple freeze / thaw cycles are not a safety issue.

I have done it. I think that it is ok as long as it is soon rather than later.