Well, Asari reproduction doesn't make sense anyway.
They reproduce purely asexually, and yet still have genetic variation. While they claim they 'scramble their DNA using their partners, but don't incorporate it', that raises the question of where the variations come from.
Random mutation doesn't occur that frequently, and is more often than not harmful, if not lethal to the organism. If Asari DNA is unstable enough to be subject to constant mutation, there would be incredible difficulty involved in producing a viable offspring, and after several generations, it wouldn't resemble anything like it's ancestors.
Mutation is counted out.
Variation could result from polyploidy, where each asari has multiple complete sets of DNA, which are randomly recombined to form the single functional set at conception, while the rest stay inactive. Humans are diploid, and have two sets, one from each parent. This is a more logical answer. Strawberries reproduce asexually *and* sexually, and are octoploids. They have 8 distinct sets of chromosomes that are randomly recombined to make unique individuals.
Asari could resemble this, where they reproduced sexually for the initial variation, then simply recombined that large pool of chromosomal material to keep up the variation during asexual reproduction. Foreign DNA would then be used as a random number generator to determine how it will be recombined.
And there is nothing about earth's classes and phyla that limit Asari, so they very well could breastfeed and reproduce asexually.
Conclusion: Asari are strawberries.