I hate to belittle the joke of this comic, but being "allergic" to something doesn't mean you react negatively to it for any reason, it means that your immune system overreacts to its admission into your body and damages you. So people actually can be allergic to certain venoms, and be killed by something that normally wouldn't kill a person. For example, it's not uncommon for people to be allergic to bee venom.
I'm not saying this situation is impossible, it's just that the joke is supposed to be that the pet he gave the guy killed him with its fatal venom, and that he just attributed it to the guy being allergic to the venom, with the punchline being that the idea's ridiculous. A similar joke would be "I poured cyanide into his coffee as a joke, but he must have been allergic to it" or "I fired a gun at him, but he must have been allergic to the bullets." If I just misinterpreted the joke, then never mind, but that's the only way I can make out the humor.