... is it the sort of thing that could then supply some functions with accessories feed/permanent live when the ignition is off?...
No. That capacitance would yield a potential of 12v for around a couple of seconds, no more, depending on the circuit in which it is fitted. In an open circuit, following a charge, it would keep its yield for weeks if not months. In a closed circuit, it would expire its charge quite quickly, as I mentioned.
It might look like it would need more than one connection, and capacitors always do, but as fitted in the picture the body of the vehicle is providing the -ve while the single white lead is taking the +ve potential, thereby creating a suppressor effect.
As martin inferred, it will be there to protect against spikes when whatever circuit it is part of is turned on, or off. If in fact it turns out to be part of an operational circuit (ie, always on during engine running) then it could well be protecting against unsolicited peaks and troughs.
Don't hold your breath awaiting a reply from Facon who will simply be suppliers or rebranders of the component, and yes I agree - it would appear to be easily replaceable as it probably blows (goes to open circuit contingency) after a prolonged period of use.