You say the fridge is plugged into a power strip and the power strip is then plugged in through a portable GFCI receptacle. If you are getting shocked by the fridge, but the GFCI is not tripping, then the GFCI is faulty and not providing you with any protection.
A plug-in GFCI receptacle tester has a button to trip the GFCI. I hope the tester you got is this kind; if so, you can use this to confirm your GFCI is busted, but I would just replace it rather than go without protection.
Now, on to the fridge. The tool you want to use here is a multimeter.
(Aside: If you're doing anything with electricity, you should have a multimeter. In a pinch, you can get by with a so-called "neon circuit tester": this has two leads and a light on the body. The light lights up when it detects a high enough voltage traveling across the leads. Be careful; many of these are intended for automobile use and will not survive mains power. I've used a couple, and I like the Gardner Bender GET-213HV, because the leads clip to the body at the right width for a US receptacle and the light is good and bright.)
Plug a 3-prong extension cord into a grounded outlet. Run the cord over by the fridge. Take your multimeter/neon tester and stick the black COM lead into the grounded U-slot on the extension cord. Touch the other lead to one of the places you felt a shock on the fridge. If it reads a voltage difference, congratulations: Your fridge is live and ready to shock the daylights out of you. Unplug the fridge and move your chilled goods elsewhere.
I don't know whether an electrified fridge could be salvaged, or whether you'll have to get a new one. Good luck.
P.S. Those receptacle testers are apparently considered somewhat unreliable. You can follow up by testing everything they test using your multimeter/neon circuit tester. Reversed polarity: one probe in the long slot, one in the ground slot, if you have power, oops. Ground: one probe in the hot slot, one in the ground slot, no power, no ground. (No ground, and you can't really test for reversed polarity, either; use the extension cord trick to get a known good ground.)
There's also always the open up the outlet, pull everything out, and see how it's wired together trick. It takes far more time, but it lets you see directly whether there's a ground screw, whether anything's attached to the ground screw, whether the polarity's reversed, etc. But don't open things up unless you're comfortable putting them back together.