I had the same feeling while watching these animes. both have dark and slow moving stories. While Hitsuji no Uta had kind of like a vampire mystery feel to it, Cossette had this gothic and ghost mystery feel. great animes for the goths and vampire lovers out there.
Both OVAs are as much about the taboo relationship between a brother and sister as they are about the gradually revealed dark secrets hidden in their pasts. Hitsuji no Uta doesn't go quite as strongly for incest angle as Angel Sanctuary does, but the subtexts are still definitely there.
Both Shiki and Hitsuji no Uta contain intricat plots involving vampires. Instead of the run-of-the-mill KILL THEM ALL concept, both shows bring forth different spins, bringing a different respect to vampire-based anime. Granted, both of these shows have entirely different plots (Shiki focusing on vampires trying to take over a vilage and the morality of murder, where as Hitsuji no Uta focuses on the concept of vampirism being a hereditary illness), the two share common elements in their ability to create a tense, eerie atmosphere and possess a story told in a serious tone.
Both are very dialogue heavy vampire stories with a minimal budget and soundtrack. While Hitsuji no Uta is definitely the better of the two, personally I give a tip of the hat to Legend of Duo for having a more interesting story.
These two are similar mostly because they both deal with individuals who have rather odd "latent" abilities (dealing with vampiric practices sort of) until arriving into their high school lives. They both also deal with blood sucking environments (vampires again). Each series has a dark undertone (Tsukihime is somewhat on the lighter side though).