The reason people in ME feel the kid is genetically there's is merely a matter of the drastically different biology and cultural differences. It's the same reason we all associate the Asari as being female even though they don't actually have a concept of gender within their own society. We look at them from the perspective of our own species. For us to have a child with an Asari we just associate it with being genetically ours because that is how having kids works for us. It is how we perceive it. Which is fine, I don't care how people view their Asari kids, I was only speaking from a technical, scientific perspective. It is fact, in ME, that no DNA is taken from the father. It is not genetically theirs, it is always the mother's. How they choose to feel about it is an entirely different matter.
There's perhaps a big philosophical debate in there somewhere about what makes the kid yours, but thats not a subject I wanna touch. None of the DNA is that of the fathers, period. Even if the randomized Asari dna could give the offspring the father's eyes or nose, it still technically isn't the fathers DNA. For some reason this reminds of Miranda and her situation of being 'birthed' by solely her father's DNA. It wasn't entirely his DNA, since she isn't a clone, but it was still based around a modified version of it. Does that make her his daughter or an imperfect clone? Dun dun dun.
It's still speculation, though. All we know for certain is that they don't use the DNA of the father and that its always randomized from the mother. We're assuming this could mean Salarian-Asari offspring could have wider eyes and narrow face. We're assuming krogan offspring have tougher skin or ridges. Its never indicated to be that way in the lore.
The only instance where we've seen that an Asari resembles a parent is with Morinth. Which means little since she looks like the mother, whom is the only one who provided any DNA for the birth in the first place. Unless we see otherwise I'm not convince the father plays any role in appearances. Infact you could argue that there is more circumstantial evidence pointing to this definitely not being the case when you factor in all the different ways the Asari have had children where physical traits COULDN'T pass down to offspring (hanar, elcor, raditation).
They would clearly have a concept of gender, whether or not there are two distinct sexual identities within their own species. I assume (which is again an assumption but it's a reasonable one considering the variety of life on Earth already) not all life on Thessia shares Asari reproductive traits, so they must have encountered other animals that do show sexual dimorphism, and therefore have an understanding of what sexual dimorphism entails. It's also shown that Asari recognise alternative sexes in other species, too (which you could argue was learned later in space-flight, but I digress), even going so far to prefer one over the other. Aria has an obvious preference for women, considering the two Omega kisses for ManShep and FemShep, for example.
There's also an obvious separation of father and mother as gendered words, defining sexual roles, which would result in a cultural separation of those identities. The father would still fulfill a "male" role, even though they have no term for that, in that they don't give birth to the child. Aethyta chastises Shep for assuming that all Asari are mothers because all Asari fulfill the biological role of sexual females, and she says that she's clearly the father regardless of that. Cultural gender.
tbh this is a complicated argument I don't care to get into, and it all depends on your definition of gender and whether or not you're referring to cultural identity, societal roles, or strict sexual dimorphism. But, yeah, they obviously know what gender is, both physically and culturally. They might not have sexual males, but they absolutely have gender.
And yes, none of the DNA comes from the father, but the child's DNA is nonetheless shaped by the father, so it's a purely down to technicality of reference with no practical difference between the two. Asari can't reproduce asexuality, so the father is still genetically required, the same as fathers are needed with our species, it's just the father has an alternative involvement that doesn't involve strict combining of genes to make a total whole.
There are plenty of ways Hanar/Elcor/etc. could factor into appearance and physiological traits. Lung capacity, vocal tone, physical strength, muscle density, weight, etc.... It's not all facial features. Those were just examples.
Asari children are 100% Asari, with all of their genetic code inherited from their mothers. The role of the 'father' is just in providing a genetic map that somehow helps randomize genetic sequences inherited from the mother to create a unique individual rather than a clone.
Which is functionally the same thing as our methods of reproduction.