I think 0V was meant to indicate zero volts (i.e. short to ground).
FWIW, I'd agree with the comments to date. The side marker connection from the converter's socket goes to ground when the lights are on, and the marker thatlights should be supplied by a relay-isolated circuit that has a permanent 12v feed and uses that ground to activate.
I'm not sure where you've found your relay, but it is a converter item, not a Fiat item (and is often, but not always, part of a Schaudt AD01 adapter when Schaudt electrics (eg EBL) are used).
Whilst not entirely conclusive, the symptoms are indicative of the wire between the converters socket and the relay or its mount having a short to earth (nipped/chafed somewhere).