Karpyshyn should probably stick to lore, or do some studying on speciation before speculating. I'll lightly cover some of the basics here from my (light) background interest in palaeontology, but your best bet for in-depth would be a biology person like @gammaraydeath or someone else involved with evolutionary biology. Everything else is just math and physics/chemistry.
We'll go with space-based hazards first, since that's why I was tagged.
The radiation is the biggest problem. Judging by how Shepard maps out mineral structures in ME1, it's common for people to just land on a planet, look for ores, and start digging. The ubiquitous mining tunnels point at this being the primary way everyone looks for resources that they can't detect from orbit. The presence of wIldcat miners like the ones found on Presrop means it can be a rather lucrative business.
Judging by how often people complain about managing extremely hazardous substances like eezo, it can be assumed that galactic standards are a bit lax with other substances. Workers are going to be contaminated; the more off-the-books you are, the more contamination you are likely to experience. Standard environmental suits are apparently terrible at managing radiation exposure judging by how quickly Ryder in Andromeda will succumb to external hazards. Full armor systems aren't even sealed effectively against airborne hazards (Shepard happily snorts Minagen X3 in Samara's recruitment mission despite being in "sealed" armor).
The prefab colonies you find on different planets are particularly bad. All that uranium sitting beneath the surface has a decay chain that includes radon, a rather hard to control element that you'll hear people discuss building up in residences and office buildings. It likes to percolate through gaps, but is limited by sealed surfaces like those you find in mines. Radon itself is bad when it decays, but its polonium decay product is worse. Then you have extra lead in your system on top of the radiation damage, so enjoy the heavy metal poisoning.
Stellar radiation is another concern. Tali's recruitment mission in Haestrom indicates that armors are weak against direct beam radiation, and colonists will have to contend with regular exposure to both ionizing and non-ionizing radiation.
Cancers will likely increase dramatically, and skin tones will probably darken over time in human populations as melanin production becomes more important.
Habitable planets, notably garden worlds, are noted in the game as highly prized because of the colonists' ability to live outside without environmental suits. However, these planets are unlikely to have the 78:21:1 ratio our atmosphere has of N:O:trace. Lower oxygen levels will require an expansion in chest cavities over time to properly oxygenate the blood, or the body will reduce functions to ensure survival. Higher levels release free radicals that severely damage organs, and people would have to adapt to that. Over time, a low oxygen population would adapt to be very dissimilar to a high oxygen population. Other gases in the atmosphere would cause a host of other issues that would have to be adapted for.
Another issue with Karpyshyn's idea is that that's just not how evolutionary biology or speciation works. When life gains access to a new niche, you get an adaptive radiation. There's tons of new niches to fill and compete for, and different adaptations occur rapidly to allow all resources to be exploited. New Zealand's avian species are a great example: there are no native non-bat mammals on the islands. Some marine life shows up on shore, but nothing moves inland. Birds filled those niches, and you can directly observe the birds behaving in the ways that mammals do in other parts of the world.
These adaptations, genetically, take a little time. But structural changes occur quickly. Modern humans aren't that genetically dissimilar from our ancestors 5000 years ago, but our bodies are building up differences. If you had/have crooked teeth, it's not something your ancient ancestors had! The jaw develops due to stress loads, so our modern soft diets have causes our jaws to soften and teeth to grow crooked. Hunter-gatherers in the past had more aligned teeth due to the need to regularly chew harder materials for nutritional sources. Your grandparents and great-grandparents likely had longer feet and toes that modern regular shoe-wearing has reduced due to growth restrictions. Even your brain is different compared to past peoples, with some speculation that modern brains tend to be good at specialization, while earlier peoples' brains were more jack-of-all-trades in function.
It's not just humans. During the end of the Pleistocene, short-toothed mammals largely vanished from the Americas. Vegetation experienced a rapid increase in silicate uptake, making it coarse and damaging to teeth. Mammals that didn't have constant tooth growth had their teeth wear away quickly, and they starved to death. Long-toothed mammals had their teeth constantly grow and replace lost material, allowing them to thrive. Ever wonder why squirrels constantly chew on things? It's directly related to these adaptations.
"What does this have to do with colonies?" you might ask. Well, planets are different! They have different surface gravities, different radiation rates, different foods, etc. The people living on those planets will change over time, even with new genes entering the pool, and different colonies will diversify from each other.
The last issue is a lore one: there simply hasn't been enough time for different species to restrict their gene pools, and their initial populations prior to space travel likely means that there won't be a bottleneck that forces them to interbreed enough to make a genetic difference.
Mass Effect: Ascension takes place in 2183, just 157 years from now. 157 years ago was the year 1869. Do you want to take a guess at how much humans have changed genetically since 1869? (hint: it's basically zero) Even with rapid expansion of international trade and travel over the past century, children born to mixed-race parents are still a minority. Here in the U.S., where multiracial relationships are somewhat common (about 15% of new marriages in 2010), only about 10% identify as multiracial. This includes cities where multiracial relationships are more likely to happen. In more rural areas, you're going to see a substantially reduced number of multiracial families.
The colonies will act primarily as "rural" environments for different species. Even with a multiracial starter group, it's going to take several generations to clear off qualities like "blonde hair" and "blue eyes". Hell, my own green eyes are considered the rarest of the common eye colors, but they're still common enough to not warrant comment.
We can't be too anthropocentric on this though, can we? The whole point was to compare humans to other species.
Asari:
This is the only species that I'm going to shrug my shoulders at, and it's because they don't mate the same way as other species. The child's genes deviate from the parent by chosen randomization for the "best qualities" of the asari's partner. I don't know anything about how this works, and neither do the writers, because they didn't really elaborate.
I do know about the New Mexico whiptail (Aspidoscelis neomexicanus) though. It's an all-female species of lizards that parthenogenetic. The parent duplicates her chromosomes, and lays clonal eggs. The offspring still have heterozygous chromosomes, and do experience genetic change over time due to mutation. So, while they all look similar, they are still rather genetically diverse.
The one issue I will put my foot down on is that the asari would have limited genetic diversity simply due to interbreeding. They achieved FTL space travel in 580 BCE, just under 3000 years ago. Sure, that sounds like a long time, but asari live for over 1000 years. Even if every generation immediately popped out a kid at 300 years of age, that's still only 10 generations. It would be the equivalent of remarking how humans have changed since 1826: ridiculous.
Salarians:
This is the one species that I can say might experience some genetic issues. 270-some generations from 520 BCE to 2183 CE (assuming 10 years for fertility) would be the equivalent of 5400 years for humans. There's a lot of time for interbreeding in this environment.
However, salarians are shown several times in the series to be highly selective when it comes to breeding. They keep family logs like horse breeders, and their background would suggest they understand the importance of genetic diversity.
If you include the lystheni cut content, it appears that salarians may have, in fact, genetically diversified in the past five millenia.
Turians:
This one is rather simple, but I want to make a note at the end.
The turians use their military monoculture to ensure that the inhabitants of different worlds don't diversify enough that a second Unification War happens. The emphasis on meritocracy additionally means that different turians will look at an individual's capability rather than their racial-equivalent history.
The note I want to make here is about their home world, Palaven. It's stated in-game that turians have thulium in their plates that help mitigate harmful stellar radiation. Thulium is rare. 200 parts-per-trillion rare. It's an r-type element, produced only in supernovae and kilonovae. It likely even rarer in the Apien Crest, as that is closer to the outer reaches of the galaxy (see a similar situation concerning the quarians here). Even if the chemistry of Palaven encourages biologically exploitable ores of Thulium, there probably isn't enough of it to really make a difference.
On other planets in the region? Forget it. The turians either have species-wide declining thulium levels in their bodies, or off-world members of the species have almost none, and possibly can't stay on Palaven long. (Hey, maybe that armor they all wear has a purpose?)
Others:
Elcor, hanar, and volus all have such specific colonization environments that you can effectively treat any world as an island situation here on Earth. A colony will genetically diversify and speciate over time based on distance. The drell are in a genetic bottleneck, and most probably look like each other.
The krogan are currently being genetically managed by the genophage. I can't say how they'll diversify, but their hardiness indicates they can survive most environments without adaptation. They do rapidly mutate though, so who knows.
Anyway, my apologies to Kahlee, but she's not going to notice the homogenization of human races.
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