The assembly is probably the same. Generally the difference between supercapacitor and battery is that charge storage in supercapacitor is capacitive (non-faradaic) whereas in battery it is faradaic, i.e., accompanied by electrochemical reactions. However the boundary can get blurred, and another "practical" distinction is that supercapacitors are tailored to high rate (high power), batteries are tailored to high capacity (energy density).
To add up in the previous answer, the device related to battery cell consists of the electrode material of battery type. For example NMC as cathode and graphite as anode for the case of li ion battery, whereas the supercapacitor cell (commercial) consists activated carbon as both cathode and anode treated as positive and negative electrode as the charge storage mechanism is through the accumulation of charges at the electrode-electrolyte interphase depending upon the polarity of electrode either positive or negative.
The assembly, in a small R&D laboratory, can be, nearly, the same.
However, the force exerted for a typical battery coin cell components is inadequate[1], resulting, usually, in erroneous measurements of electrode performance and (a higher) ESR (Equivalent Series Resistance)[1].
1. Using coin cells for ultracapacitor electrode material testing https://utw10193.utweb.utexas.edu/Archive/RuoffsPDFs/258.pdf