The answer seems to be that it is a misnomer or at least bad choice of name. They talk about (lithium) ionic superconductors at each occurence of the term superconductor which makes me think that they actually mean materials that only let lithium ions through but not others (because this is how their device actually works).
There are many criteria by which superconductors are classified.
Superconductor material classes include chemical elements (e.g. mercury or lead), alloys (such as niobium–titanium, germanium–niobium, and niobium nitride), ceramics (YBCO and magnesium diboride), superconducting pnictides (like fluorine-doped LaOFeAs) or organic superconductors (fullerenes and carbon nanotubes; though perhaps these examples should be included among the chemical elements, as they are composed entirely of carbon).[12][13]
