Well, that answer from u/DaphneDK gives you the basics and directs you to the places I went for more information too. Justin's Epitome of Pomepeius Trogus and Appian's Syrian Wars contain the same basic story about Seleucus I Nicator. First Justin:

His mother Laodice, being married to Antiochus, a man of eminence among Philip’s generals, seemed to herself, in a dream, to have conceived from a union with Apollo, and, after becoming pregnant, to have received from him, as a reward for her compliance, a ring, on the stone of which was engraved an anchor and which she was desired to give to the child that she should bring forth. A ring similarly engraved, which was found the next day in the bed, and the figure of an anchor, which was visible on the thigh of Seleucus when he was born, made this dream extremely remarkable.

This ring Laodice gave to Seleucus, when he was going with Alexander to the Persian war, informing him, at the same time, of his paternity. After the death of Alexander, having secured dominion in the east, he built a city, where he established a memorial of his two-fold origin; for he called the city Antioch from the name of his father Antiochus, and consecrated the plains near the city to Apollo. This mark of his paternity continued also among his descendants; for his sons and grandsons had an anchor on their thigh, as a natural proof of their extraction.

And now Appian, whose story contains the additional detail about finding an anchor near Babylon:

...also that his mother saw in a dream that whatever ring she found she should give him to carry, and that he should be king at the place where he should lose the ring. She did find an iron ring with an anchor engraved on it, and he lost it near the Euphrates. It is said that at a later period, when he was returning to recover Babylon, he stumbled against a stone and that when he caused this stone to be dug up an anchor was found under it. When the soothsayers were alarmed at this prodigy, thinking that it portended delay, Ptolemy, the son of Lagus, who accompanied the expedition, said that an anchor was a sign of safety, not of delay. For this reason Seleucus, when he became king, used an engraved anchor for his signet-ring.

So the origin of the anchor symbol might originate with a ring given to Seleucus by his mother. The way Appian describes it seems like he had a similar ring recreated when he became king to use as his signet. Alternatively, there was never an original ring, and he adopted the symbol as his signet following later events.

There's probably also some truth to the birthmark story. I know the answer you linked dismisses that as suspiciously close to Alexander, but I'm both struggling to find any story about Alexander having a particular birthmark, and of the oppinion that a hereditary birthmark is a very peculiar thing to just make up. A skin blemish like that, passed from generation to generation is extremely rare, but not unheard of, and won't appear on every child. The birthmark's existence seems plausible given Justin's/Trogus' description.

Of course, it's the story of Seleucus' divine parentage that gets into the clearly fabricated and parallel to Alexander details. Alexander the Great had claimed to be the son of Zeus, king of the gods, himself. A story was propagated that Seleucus was the son of Apollo. This lent the Seleucids the same perceived divine approval that Alexander the Great claimed. It also made Seleucus himself seem that much more impressive and divinely chosen to rebuild Alexander's empire. Incorporating a well known and oddly shaped mark on his body with Apollo was an easy way to "prove" his divine parentage.

From there, you've got king who claims his divine father marked his body with a vaguely anchor-shaped mark, who just happened to have been a Ptolemaic admiral before conquering on his own. It seems like an anchor would be an easy choice. It's associated with divine parentage and a projection of naval prowess. If the symbol was not adopted by Seleucus himself until after securing some Mediterranean coast and pushing the Antigonids out of Anatolian territory then it may have commeorated those victories. Also, if it really was a hereditary mark, later generations would have considered it a symbol of legitimacy.

Finally, the anchor's most enduring use by the Seleucids was as a symbol on coinage. The coins of most prominent Greek cities at this point would usually mark one side of the coins with their local emblem, often something to do with the foundation of the city. Seleucus was founding something new, and establishing a new Greek tradition in the east. He was the founder and selected a symbol to reference his own foundation. It seems like an anchor would be a prominent choice to include in his iconography, if even one or two of the above stories was correct.