"Before our study, people always assumed that permanent magnets could only be made from solids."
If you finely divide a magnet into tiny magnets and suspend these particles in a liquid, is that really a liquid magnet, or just a bunch of solid magnets floating in a liquid?
You've actually stumbled upon the central premise of this discovery.
If you finely divide a magnet into tiny magnets and suspend the particles in a liquid, you get... a normal ferrofluid. It's magnetic in the presence of an external field, but will lose its magnetization once that field is removed. This is because permanent magnetism (i.e. ferromagnetism) is a bulk property.
Every atom has a magnetic moment, but they are normally randomly aligned and thus the macroscopic field cancels out.
It's only when these moments are aligned that a macroscopic magnetic field arises. Permanent magnets have the requisite crystalline structure for this to happen. As you chop it up, this macroscopic organization is destroyed.
If you get into the details, what they've done is to use the surface tension of the oil-in-water to "jam" an outer layer of magnetic particles and prevent them from rotating, thus preserving their magnetic alignment. This, in turn, is apparently enough to keep the free-floating, unjammed particles inside the droplet aligned as well, thus turning the entire droplet into a magnet. Pretty interesting, because without the membrane, none of the ferrofluid is magnetic, but with the membrane, all of it is.
The physics-y way to answer this question is usually to define things in terms of the scales at which it's probed. If everything you can sense with macroscopic instruments is consistent with the liquid properties and the magnetic properties, it's a magnetic liquid. That it's "really" just solid magnets suspended in a liquid seems not much more relevant than the fact that solids are "really" fluids if you watch them on the astronomically long timescales on which they lose solid form due to tunneling.
In this case, it's a liquid magnet down to roughly the scales of the nanoparticles, which is only a bit bigger than the atomic scale on which nothing is a normal liquid.
I think the question doesn't make sense, because if you take it further something interesting appears. If you do have tiny solid magnets in a liquid, why don't they rotate and cancel out?
They clearly cannot do so because macro sample would lose its properties. It simply cannot be tiny magnets suspended in a liquid. You gotta provide finer definition of a liquid because it typically does not have a structure. In your example there are constraints to movement and position of tiny magnets.
It's a magnet membrane surrounding a portion of liquid.
On one side your right, this isn't really a "liquid magnet", but on the other, this behaves like a magnet while retaining many of the properties of a liquid.
As I understand it (which, like most of the population of the planet, is not very much), solid magnets are also just a lot of smaller magnets floating in a macrostructure.
Permanent magnets are locked in place. If they're floating freely, the "tiny magnets" just snap together and cancel each other out. That's why heating (increasing particle mobility) a permanent magnet makes it lose its magnetization.
Not really floating, since that implies a certain freedom of movement that atoms in a solid don't have. But otherwise you are correct, since each electron is a tiny magnet (protons and neutrons are also magnets but their magnetic field strength is much smaller so they can safely be ignored). A macroscopic magnet is just a material where the electron magnetic fields add together to an macroscopically appreciable value, rather than cancelling each other out, i.e. the direction of the electron magnetic fields are largely aligned.
If you finely divide a magnet into tiny magnets and suspend these particles in a liquid, is that really a liquid magnet, or just a bunch of solid magnets floating in a liquid?