I recently came across this title (?) of Poseidon on another question asked on this sight, but have no idea what it means. I have tried a direct Google search, but come up empty handed. The only thing I found was the question I got it from.
Where does Poseidon get called the God of Kin?
Is he worshiped any differently as the God of Kin?
What does he function as when he's the God of Kin?
Presumably the other question to which you refer in your own inquiry is "Why would Aphrodite be armed?", asked by @yannis, in which he links here to the 1918 translation of Pausanias' Description of Greece 3.15.10, by William Henry Samuel Jones and Henry Ardene Omerod.
Meaning
Jones & Omerod's translation here is a bit loose. Pausanias does not really say anything that amounts to calling Poseidon the "god of" such-and-such. Rather he applies the title Genethlios (Latinised as Genethlius) to Poseidon, saying that there was a sanctuary dedicated to this god at Sparta, "not far from the theatre" and at which sanctuary there were hero-shrines to Kleodaios [Cleodaeus] (a grandson of Herakles) and Oibalos [Oebalus] (the father of King Tyndareus).
A fairly direct translation of Genethlios, via the Latin roots of English, is "Generator," in the sense of "Engenderer" or "Begetter," i.e. one who gives birth, or (since we're referring here to a male, and males typically cannot give birth) the source of one who is born. The term was even used to mean "birthday." In his book Greek Gods Abroad: Names, Natures, and Transformations, Robert Parker translates the name as "of Begetting".
As Oibalos is one of the earliest kings of Sparta—only fourth since the city's foundation by his great-grandfather Lakedaimon son of Zeus—Claude Calame finds it significant that Oibalos's hero-shrine is located in close proximity to the Poseidon Genethlios temple. In his chapter contributing to Interpretations of Greek Mythology, he understands this god's name to mean:
the guardian of the gene, the clans constituting the first Spartan citizens.
Gene here is not referring to the modern scientific term in English, although the root concept is similar. Genethlios is thus also interpreted as "Of Kindred" in the sense of belonging to a particular "race" or family; or essentially a similar type of grouping bearing some level of homogeneity.
Bithynia (A Few Days Due East of Troy)
In his Argonautika, Apollonius Rhodius says that Amykos [Amycus], the gigantic king of the Bebrykes [Bebryces] of Bithynia, who challenged foreign visitors to his land to a boxing match which they would not survive, was the son of Poseidon Genethlios by a local nymph named Melia.
In the "Explanatory Notes" to his own translation of Apollonius, entitled Jason and the Golden Fleece (The Argonautica), Richard Hunter says:
The title Genethlios ('Generative') implies that Poseidon was the creator and protector of the Bebrykians.
In Curiosities of Indo-European Tradition and Folk-Lore, Walter Keating Kelly takes a different approach to this, claiming that Genethlios is a surname which marks Poseidon
emphatically as the god at whose disposal is the moisture that is the cause of all fruitfulness and nourishment.
Because Melia's name means "ash-tree," he goes on to say that the god's connection with this nymph "distinctly represents" a certain archetypal Indo-European tree (apparently) "in which are comprised all the seeds of vegetation."
Argolis
The place at which Theseus was born in the city of Troizenos [Troezen] in Argolis, on the way down to the harbour, was called Genethlion, "Birthplace," or more literally, the name of the spot was simply just "Of Birth." (The Latinised version of the name is Genethlium.) There was a connection here to Poseidon, who not only was much revered in this city, especially at this spot, but was also reputed as the real father of Theseus, over against Aigeus [Aegeus].
At another place in Argolis, near Lerna, there was a village called Genesion [Lat. Genesium]. Pausanias reports that here, "By the sea, is a small sanctuary of Poseidon Genesios." According to William Smith's Dictionary of Greek and Roman Biography and Mythology this epithet means "Father."
The name is identical in meaning with Genethlius (γενέθλιος), under which the same god had a sanctuary at Sparta.
Possible Cultic Activity in Veneration of Poseidon Genethlios
According to the Description of Greece, the springs of the Inakhos [Inachus] River were on Mt Artemisios. These springs did not run far. The river, so long as it flowed by the road across the mountain, was the border between Argos and Mantineia, the latter of these being in Arkadia [Arcadia]. A road led over Mt Artemisios from Argos into Arkadia. After crossing into Mantineia over Mr Artemisios there was a plateau called "the Untilled Plain," which was claimed by the Arkadians [Aracdians] to be the birthplace of Poseidon.
Rainwater coming down from the mountain prevented this land’s arability, and it would have been a lake if the water did not disappear into a chasm there, after which this same water rose again at Dīnē, "Whirlpool," a freshwater stream rising out of the sea in Genethlion (Theseus' birthplace). The Argives used to cast horses adorned with bridles into Dīnē as offerings to Poseidon (perhaps = Poseidon Genethlios or Genesios).
Zeus Genethlios
Zeus, by the way, was also venerated as Genethlios. The translation from kk_Omatsu's Wattpad blog renders Genethlios as "creator of life".
In Pindar's Mythmaking: The Fourth Pythian Ode, Charles Segal says that Helios, the sun-god, too was given the title genethlios patēr (patēr = "father"),
a frequent epithet of Zeus as the source of procreative energy and the guardian of the patriarchal family.
In Vol. 1 of The Cults of the Greek States, Lewis Richard Farnell interprets Zeus Genethlios as "the god of the birthright," who could assume the function, like the Erinyes, of specially punishing "wrong done to parents" and executing the curse a father has placed upon his offspring.
A letter from Laodike [Laodice] III, Queen Consort of the Seleukid [Seleucid] Empire, to the people of the city of Sardeis [Sardis] acknowledges these people's pledge to offer a sacrifice to Zeus Genethlios for the protection of the royal family members. The letter reads as follows:
Queen Laodike to the council and people of Sardeis: greetings.
Your envoys - Metrodoros, Metrophanes, Sokrates and Herakleides - have presented your decree, in which you voted to ... perform a procession and sacrifice to Zeus Genethlios for the safety of our brother Antiokhos [Antiochus] and ourselves and our children; and the envoys addressed us in accordance with what was recorded in the decree.
We had gladly accepted the honours, and we praise the people for their eagerness. We will always try to help in establishing something beneficial for the city, and the envoys will report to you about these matters.
Year 99, 10th day of Panemos
