What is shown on the right, sticking to the metallic disk being lowered into the liquid, is magnetic soap. (On the left is normal, non-magnetic soap.) The little yellow blob that lifted with the magnet-on-a-stick through the clear liquid and then fell when the magnet was lifted right out of the liquid, is the soap itself.
A solution of ferric chloride by itself also reacts very slightly to magnets, but it finds most of its use as a coagulant, not as a soap.
This discovery isn't, however, likely to result in a bar of soap you can stick to a magnet on the wall of your bathroom.
The biggest and most immediately obvious application which was mentioned in the press release is for cleaning up oil spills. One method of combating an oil slick is to spray dispersants on the slick. This is basically soap, and in the same way dish soap isolates the grease from your dishes in little balls of soap (micelles) and prevents it from re-sticking, these oil slick dispersants break the continuous mass of oil floating at the surface into little isolated balls which are soluble in water and don't coat the surface of the sea. The idea is that by dispersing the oil slick in this way, it's not coating everything that touches the surface of the water, and it spreads out so the bacteria which eat oil can work on it.
If magnetic soap were made into such a dispersant, then after the oil slick was broken up into micelles, a magnet could attract them into a skimmer and they could be removed from the ocean entirely.
Any other application where you might want to separate the soap from the water after the soap has done what it needs to do could use this as well. Soap and cleaning applications are only one of the many uses of surfactants in industry; another common one is antifoaming agents. (Or, for household soaps, foaming agents.) Foam is really annoying in a reactor, especially when it overflows the top of the tank, but sometimes antifoaming agents could affect the process downstream, either by interfering with the reaction or by remaining there as a contaminant in the final product.
The press release also mentioned that this could lead to a soap with a "magnetic on and off switch". Since there's a big difference between a soap that is attracted to a magnet and one that can be turned on and off, I obviously had to figure out just how that could work. This is not a concept I've encountered before, possibly because I haven't done much work with surfactants.
In my searching, I found a paper about a soap that has more or less effect on the solution's surface tension based on whether or not it was exposed to ultraviolet light. The short version of how that works is that, embedded in the hydrophobic "tail" section of the surfactant molecule, there is a UV-sensitive chemical bond which twists in a different way depending on whether or not UV light hits it, and it can twist back and forth as you turn the UV light on or off. The change in shape of the "tail" then affects how the molecule forms micelles and gathers at the surface of the water, and thus affects the effect it has on the solution's surface tension.
In addition to that, returning to the ferric chloride solution having a very slight response to magnets, the response was a measurable change in surface tension as the iron ions migrated to the surface of the water closest to the magnet. Nothing visible to the eye, but measurable. So either or both of a change in the shape of the surfactant molecule or a change in location of the surfactant molecule could change the surface tension of the solution—and changing surface tension is really what surfactants are all about.