I have done a lot of reach on the favor and love of the gods for various mortal families. I have traced a number of the lineages up and down the vast genealogy tree of mortals in Greek mythology. My hope was to understand the gods' affection for various families; generation after generations. Here's what I came up with;
The love of the gods for the Trojans might simply be based on their manly beauty. It is simply a fulfillment of “Maya’s Law” which states that the Olympians have a marked preference for mating with Ionians and barbarians. There seems to be no lingering affection for other family members of Endymion (immortaized lover of Selene) or Cephalus (mortal lover of Selene's sister and grandfather of Odysseus.)
The gods short lived affection for Tantalus (and his descendants) could be contrasted with their short lived affection for Ixion. After murdering his intended father-in-law Ixion could find no eartly king or minor deity willing to purify him. Zeus to him up to Olympus to perform the ritual. In gratitude Ixion violated the Laws of Hospitality by attempting to rape Hera. The parentage of Thessalian Ixion is disputed and vague but none of his potential parents were beloved of or dear to the gods and certainly none of his children were. The only cause I can suggest for the Olympians love of the Tantalides is that Tantalus was Zeus son.
Few men and their descendants seem as dear to the gods as do the Aeacides. They too were Thessalian. For three generations they were given unheard of honors by the gods. They too are descended from Zeus.
Melampus on the other hand was not the son of Zeus nor was his father a favorite of the gods. Many of his children received divine favor. His abilities as a seer and piety seem equal to the Theban seer Tiresias but the former's family received no such benefits.
So I found no clear proof the gods can't tell individuals apart from a their fathers. Rather the gods' favor seems to fall upon handsome men, pious men, great warriors and wise seers. To honor their father's the gods sometimes confer favors onto the children. The children inherit the favors owed their fathers and maybe their reputations.
But in the end each bread-eating man stands alone before the gods.