Pull the ciggie lighter socket away from it's mount so you can measure the +12v voltage at the back of the socket with the fridge plugged in and let us know what you get. Measure the voltage of negative outer side of the socket and a decent body earth (under the dash/direct to firewall via bolt), this should be zero to less than 0.2v with the fridge connected/running. My guess is a dodgy earth.
People have OE lighter plugs in the back of their vehicles which have a longer run of marginal wire gauge that are able to take the 5 amps or so of a fridge (not an overly heavy current). Main issue with ciggie lighter plugs/sockets is mechanical (susceptible to vibration/falling out), they are after all able to take the current of a cigarette lighter (15amp)
http://www.rowand.net/shop/tech/wirecapacitychart.htm" onclick="window.open(this.href);return false;
Gauge 12V
22 5A
20 8A
18 10A
16 20A
14 40A
12 60A
10 100A
8 150A
6 ??A
4 ??A
2 ??A
1 ??A
0 ??A
http://www.tnt-audio.com/clinica/awg_e.html" onclick="window.open(this.href);return false;
So 1.8 mm2 ia about 15 AWG so should handle 30 amp for short runs and about 20 for decent runs (front to back of vehicle).
http://www.mwswire.com/barecu5.htm" onclick="window.open(this.href);return false;
Resistance of 3m (10 feet) of 15 AWG is 0.03 ohm. The maths at 5 amp drain would be 0.15V voltage drop over a one way trip = 0.3volts total for return. Add another 0.2 volts for plug losses and it's still within reason and overall nowhere near overloaded.
Practical upshot is the wire used is adequate for this purpose (not "best practice", but adequate nonetheless), and his problem lies somewhere other than with the gauge of wire used.
George Carlin, an American Comedian said; "Think of how stupid the average person is, and realise that half of them are stupider than that".