It seems like you weren't allowed to be a Greek hero unless you were in some way abandoned at birth. Oedipus is only one of many figures who were ditched in the wilderness by their parents. Heracles' mom ditched him in the woods, because she was afraid of Hera's wrath. Jason's mom sent him to be raised by Chiron the centaur, so that his evil uncle wouldn't kill him. Atalanta's dad abandoned her in the woods because she wasn't a boy.
Like Oedipus, Paris of Troy was abandoned out of fear. His mother, Hecuba, had a dream that he would one day bring about the downfall of their city, so his father, Priam, abandoned him on a mountain. Also like Oedipus, baby Paris was saved by a shepherd. Also similarly to Oedipus, baby Perseus was locked in a chest with his mother and thrown into the sea because his granddad heard a prophecy that Perseus would one day kill him. Some versions of Oedipus's story actually say that Oedipus was thrown into a chest just like Perseus.
Of course, the Greeks weren't the only ones to insist that their heroes have been abandoned at birth. The Roman culture heroes, Romulus and Remus, were tossed into a river by their evil uncle and survived by the grace of their dad, Mars, and a friendly she-wolf. The Judeo-Christian figure, Moses, was also put into a river when he was a baby. In his case, though, his mother made him a floating cradle out of rushes to protect him from the Pharaoh who wanted to kill all the male Hebrew babies. Luckily for Moses, he was found by the Pharaoh's daughter, who raised him as her own. The Persians also had a story that their emperor, Cyrus the Great, was abandoned by his grandfather, Astyages, when Astyages dreamed that Cyrus would one day take his throne. Like Oedipus, Cyrus was found by shepherds, and he eventually sat on his granddad's throne.
This popular trope from mythology pops up all through popular literature of all kinds too. Other heroic orphans include: Oliver Twist, Harry Potter, Lyra Balaqua, Hansel and Gretel, Frodo Baggins, Little Orphan Annie (obviously), Huckleberry Finn, Spiderman, Superman... We could just keep listing names, but then you'd be here all day. We wonder what it is that people seem to like so much about stories where an abandoned child grows up to do great things. Who knows? Maybe, people just like rooting for an underdog. Whatever the case, it's clear that Oedipus is in a pretty popular club.