I think more people are familiar not just with Kong but also how he lives on an island, falls for the girl, climbs up the Empire State Building. His story in itself is iconic.
If you compare it to Godzilla, people know he's a big Japanese reptile who breathes fire but less people are familiar with the specifics of the franchise. I think the 1998 film really had an impact on a lot of 90s kids (and younger) about their idea of the character before the Legendary reboot.
Many older Godzilla films being billed as afternoon TV fare probably had a role in that too.