Unique Copper Wire That Can Store Electricity

You are correct, it is not a battery. I can see it could be useful in applications which require a momentary large inrush of current such as a motor start up that a smaller battery could not deliver. I would assume the switch would be on the load side so that it would basically be a supercapacitor in parallel with the battery. It's obviously good only for conducting DC power.

As it is a linearly-distributed capacitance, I can see where this may have useful applications in pulse-formers and generators. I've used lengths of unterminated coaxial cable for this purpose in the past. Charge-up the center conductor through (typically) a resistor with the coax shield grounded and then discharge through a FET switch to the load. The discharge propagates down the coax until it reaches the end, generating a constant-current pulse meanwhile whose duration is a function of the coax length and propagation velocity. Small-signal stuff in my application, but this same approach can be scaled up by ganging sections in parallel to generate fast pulses having extremely high currents for fusion experiments, lasers, railguns, magnetisers, etc.

Really, this technology boils down to being just a new way of making coaxial cable, no? And distributed capacitors.

A very low impedance coax cable.

Yes, and according to the article, a coax wot stores not just energy, but powerful energy. Clearly they're making direct application of the Strong Nuclear Force; the author of the piece just doesn't know it yet. (psst: I won't tell if you won't ).

Seriously though, if they want to use nanostructures to pack the most energy into the smallest volume, using a cylindrical wire as the form is the wrong approach. For one thing, just one glance will tell you that it has entirely the wrong geometry. That center conductor places severe constraints on the available surface area and, if it's capacitors you want to build, you're gonna need all the surface area you can get. Tightly-wound parallel sheets would be a far better approach as any capacitor manufacturer would tell them. Even better, some kind of porous approach akin to solid tantalum capacitors. What they've come up with is basically a way of using nanostructures as a capacitor dielectric but nothing else about it is really all that new - coaxial cables have been around since Christ was a colonel.

Hard to say what the impedance is. I suspect you're right but there is an (unspecified) inductive component as well as a finite propagation velocity that will also influence the effective impedance.