Nintendo kind of tried that with "friend codes" but they were just annoying and spread online anyway, which is kind of fair since today it's not uncommon for people to be friends with people in distant places and the internet should allow us to be able to connect and play with each other no matter where we are.

The original solution to just playing with local friends was LAN parties, but private servers were great too where they were actually private/hosted entirely by the players and not just part of some corporation's online platform.