I think the common factor is that if any monster in the formation is immune to instant death, Odin will always fail. Try finding a battle with one monster immune to instant death, and another not. Try summoning Odin when both are alive, and the one immune to instant death is dead. See what happens.
I've always believed if any monster is immune to death it will automatically trigger a fail.
Well, there are some simple documented behaviors/rules that I should write a separate post about (feel free to split this thread if neccessary).
1) If you make a multi-status attack (a physical attack+statuses or a special attack) such as Bad Breath (Marlboro), protection against one status that Bad Breath tries to inflict means you are immune to the entire attack.
For the next rule, it depends if you are using the Advance version or the SNES version.
2A = FF4A, 2B = FF4 SNES
2A) if you make a multitargeted status attack such as Piggy, it doesn't matter if it is successful on every target, even if one is immune.
2B) If you make a multitargeted status attack such as Piggy, it has to
be successful on every target, even if one character is immune to the status, otherwise it will fail. This is why you can see certain lunar overworld formations like those squidish monsters can't be petrified, but the rest that are not immune (the puddings, bomb, and the virus monsters) die promptly.
Simple rules eh?... unless you want it refined...
Odin follows rule 2B, but it isn't a death status attack at all...