He has incredible powers and strength, and out of all the many ways villains try to hurt him, they always fail because in the end, what can hurt him is a piece of a rock. Batman, as Bruce Wayne, has powerful lineage, just the Greek heroes. He may not be descended from the gods themselves, but there still exists a comparable mythology around him. His parents, but particularly his father, was a very rich and powerful man. His parents were philanthropists and very prominent in the world of business, of which Bruce Wayne is the heir of their legacy. His parents' souls exist in heaven, or in the clouds, or somewhere where Bruce can only return to them through death, which is similar to Greek mythological heroes that can return to or greet their god-ancestors after death and after living a life truly reflecting heroism. Superman's father...

He also came from a strong family and cannot return to them. The deaths of these modern heroes' parents distance them from their "god-like" parents (in reputation and deeds) just as human and demi-god Greek heroes from mythology are distanced from their ancestors who are gods and other kinds of powerful mythological beings who endow them with strength and serve as motivation.
The examples mentioned here are just a few of the examples that are influenced by the Greek concepts of the hero. Of course there are modern heroes that have little to no relation to Greek hero stories, but the overall point is that there are heroes of the modern day that do keep these stories alive in some way, but emulated the character structure, myth structure, and overall mythology of the modern hero.