I'm willing to try cleaning with Borax, BUT...
Boric acid and Borax are not at all the same thing.
Boric acid is a boron atom bonded to three (-OH) groups.
Borax is a ionically bound salt that contains sodium.
I believe the "active ingredient" in both is the boron that may have insecticidal properties just as both copper, arsenic and zinc have biocidal properties. However, I have trouble with the notion that Boric Acid or Borax would be more effective in killing bacteria (which are most commonly the source of foul smells) than bleach. I say that because a quick Google search will reveal that Boric acid and Borax are most commonly used to rid houses of insects like ants, silverfish and cockroaches. If either also worked well to remove odors from old fridges, it would be common knowledge, and it wouldn't be hard to find any reference to that application on the internet. Instead, you find nothing on the internet about that.
I'm willing to try cleaning a fridge with Borax or Boric Acid, whichever is recommended, but I remain skeptical. I do know, however, that borates are very effective against mold and fungii, so I'd be open to the idea of cleaning mildew off walls or grout in a bathroom with Borax or Boric Acid. However, mold and fungii belong to a totally different genus than bacteria, so what may work well to kill molds and fungii might not do anything at all to bacteria, and vice versa. For example, borax is even used as a food additive in some countries because of it's low toxicity. But, it's highly effective against some insects and most molds and fungii.