No, Narcissus has a few different stories. There's Narcissus and Echo, Narcissus and Ameinias, Narcissus and Tireseas... You can argue that Tireseas is just foreshadowing for later on, but Ameinias isn't even mentioned by name in Ovid's version, which focuses more on Echo.

I think the key distinction between the story of Narcissus and modern cultural norms is that Narcissus' self-obsession harmed only himself. He started to death because he couldn't look away from the mirror pool. Whereas today, people imagine that narcissistic people are cruel abusers.

A much better mythological parallel is Theseus. Theseus abandoned Ariadne after she helped him, and kidnapped 12-year-old Helen of Troy to be his wife. Theseus was abusive and pedophilic and it was all about his ego. He did that stuff for personal glory.

So Narcissism is when your pride hurts yourself. Theseism is when your pride hurts others.

And if you think "narcissists" are abusers, I can tell you've just fallen for Cheistopher Lasch's ableist propaganda, because there's nothing in Greek mythology to support that.