No, "allele" is not the word we want, though we should be using it in preference to "gene". "Allele" just means "a particular variant of a gene". Technically speaking, "gene" means all the ways of coding for some particular set of structures (or rather the proteins that end up constructing them, or otherwise affect development). For example, humans have two primary genes for blood type. The first gene determines the Rh factor, with one allele of that gene coding for positive Rh, and the other for negative Rh. The second gene determines the ABO encoding, with one allele coding for O, a second for A, and a third for B. And of course, the alleles on each copy of the gene combine to produce different phenotypes, which can often be simplified to the "dominant recessive" model when there are only two common alleles in a population (e.g. Rh). ABO typing is more complicated -- A and B refer to types of "antigens" (surface markers) that your blood cells may have. Each is produced if you have at least one allele of that type. O, in contrast produces no antigens. (There are actually a whole passel of other genes that code for existence of a whole lot of other antigens and typing factors, but the variants are a lot rarer, so most people don't need to worry about them.)
The term wedifrid is asking for (and that I would really like to have) is about the frequencies of alleles. There is casual talk of someone's son being 50% related to his father. Certainly exactly 50% of his alleles were copied from his father. On the other hand, we should say that he's also 50% related to his father's identical twin brother, where there is no direct copying -- just the happenstance that this set of 50% of alleles is identical to that of his father's identical twin. But, as it turns out, of the 50% of genes that weren't copied, a very very high proportion will be the same as in his father (or indeed his uncle).
A natural distance to define on these sets of alleles is the l_1 distance "how many are different". (we can choose this on the level of DNA letters, codons, codons with equivalents lumped together, or "expresses same protein"; and measure slightly different things). For most genes there is effectively only one allele, so this measure of similarity doesn't go from 0% to 100%. Instead, it varies in a tiny fraction of 1% around 100%. If we do want it to be able to drop down to 0%, then we shouldn't count those genes with only one allele. How about genes with two alleles, but one is extremely rare? Perhaps entropy of each gene would be a reasonable weighting. Just counting "is there a path of descent to a common ancestor" certainly won't let us stretch the scale out to 0% either, because in that sense we're all heavily related. We want some (hopefully mathematically formalizable) sense of "this guy's deviance from the human average overlaps by x% with this other guy's deviance from the human average". Or at least a more precise word for that, rather than just talking about shared gene (allele) percentages, which will be extremely close to 100%.
Daniel Dennett has advanced the opinion that the evolutionary purpose of the cuteness response in humans is to make us respond positively to babies. This does seem plausible. Babies are pretty cute, after all. It's a tempting explanation.
Here is one of the cutest baby pictures I found on a Google search.
And this is a bunny.
Correct me if I'm wrong, but the bunny is about 75,119 times cuter than the baby.
Now, bunnies are not evolutionarily important for humans to like and want to nurture. In fact, bunnies are edible. By rights, my evolutionary response to the bunny should be "mmm, needs a sprig of rosemary and thirty minutes on a spit". But instead, that bunny - and not the baby or any other baby I've seen - strikes the epicenter of my cuteness response, and being more baby-like along any dimension would not improve the bunny. It would not look better bald. It would not be improved with little round humanlike ears. It would not be more precious with thumbs, easier to love if it had no tail, more adorable if it were enlarged to weigh about seven pounds.
If "awwww" is a response designed to make me love human babies and everything else that makes me go "awwww" is a mere side effect of that engineered reaction, it is drastically misaimed. Other responses for which we have similar evolutionary psychology explanations don't seem badly targeted in this way. If they miss their supposed objects at all, at least it's not in most people. (Furries, for instance, exist, but they're not a common variation on human sexual interest - the most generally applicable superstimuli for sexiness look like at-least-superficially healthy, mature humans with prominent human sexual characteristics.) We've invested enough energy into transforming our food landscape that we can happily eat virtual poison, but that's a departure from the ancestral environment - bunnies? All natural, every whisker.1
It is embarrassingly easy to come up with evolutionary psychology stories to explain little segments of data and have it sound good to a surface understanding of how evolution works. Why are babies cute? They have to be, so we'll take care of them. And then someone with a slightly better cause and effect understanding turns it right-side-up, as Dennett has, and then it sounds really clever. You can have this entire conversation without mentioning bunnies (or kittens or jerboas or any other adorable thing). But by excluding those items from a discussion that is, ostensibly, about cuteness, you do not have a hypothesis that actually fits all of the data - only the data that seems relevant to the answer that presents itself immediately.
Evo-psych explanations are tempting even when they're cheaply wrong, because the knowledge you need to construct ones that sound good to the educated is itself not cheap at all. You have to know lots of stuff about what "motivates" evolutionary changes, reject group selection, understand that the brain is just an organ, dispel the illusion of little XML tags attached to objects in the world calling them "cute" or "pretty" or anything else - but you also have to account for a decent proportion of the facts to not be steering completely left of reality.
Humans are frickin' complicated beasties. It's a hard, hard job to model us in a way that says anything useful without contradicting information we have about ourselves. But that's no excuse for abandoning the task. What causes the cuteness response? Why is that bunny so outrageously adorable? Why are babies, well, pretty cute? I don't know - but I'm pretty sure it's not the cheap reason, because evolution doesn't want me to nurture bunnies. Inasmuch as it wants me to react to bunnies, it wants me to eat them, or at least be motivated to keep them away from my salad fixings.
1It is possible that the bunny depicted is a domestic specimen, but it doesn't look like it to me. In any event, I chose it for being a really great example; there are many decidedly wild animals that are also cuter than cute human babies.