By this point in Leftover Soup, I hope it's obvious that I wrote Max and Lily as each other's counterparts. Lily is Bizarro Max, or possibly vice versa.
The parallels between them are more than superficial. It's not just that they're both under five feet tall, that they both have eyebrows that don't match their hair, that one is frequently grumpy while the other is cheerful, or that one loves dick and the other hates it. It's more than that.
Both characters have unique, unusual philosophies; philosophies that are so intense, so fanatical, so behaviour-defining, and so far off from societal norms that Max and Lily can only exist because their friends bend over backwards to accomodate them - and both of these philosophies stem from the same fundamental a priori principle: that, intelligence being equal, there is (or there should be) no ethical distinction between human beings and animals.
...Jeez, that was a heck of a sentence.
Anyway, in Lily, this principle results in the degradation of the human. Lily regards women with contempt and men with disgust because she views us as indistinguishable from bonobos with a bipedal stance and delusions of civilization.
Conversely, in Max, this principle results in the elevation of the animal. Her concern for animals (as manifested most noticeably in her veganism) stems from the fact that she considers there to be no ethical difference between a cow and a human being with a cow's level of cognition. Only the mind is important - to discriminate based on shape and genome is wrong.
(No doubt many of you are now wondering what Max's position on bestiality is. Max, of course, is against bestiality for the same reason she is against statutory rape - inability to meaningfully consent. If a quadruped could be cognitively elevated to true sapience, as with the talking cow example in the comic, she wouldn't consider sex with it to be bestiality.)
Both characters exemplify the inherent danger in establishing a reasonable-sounding set of first principles and then deriving a philosophical system from them without regard for practicality or the pre-existing culture. Their mindsets, like the systems of many of the more unique philosophers before them, are like two different flavours of nonEuclidian geometry - internally consistent, but, at times, surreal.