18,446,744,073,709,551,616 is 2^64. I assume that this means the planet generation algorithm is based on a random seed that is a 64-bit number (e.g. the long type in many programming language). I don't know how (or even if; how would anyone check?) they guarantee that all possible inputs are used and that none are repeated.

EDIT: While it's still true that this number is 2^64 (and it still appears in some statements in the lore, so could be said to be the number of planets that exist in-universe), there are not this many planets in the game. Since the Atlas Rises update, there are 256 (257? It's not entirely clear) galaxies, and each planet has a 12-digit hex coordinate that identifies it within the galaxy. This coordinate is divided into parts:

• 3 digits ea. X and Z coordinates of region (a region is a cube-shaped volume of space)
• 2 digits Y coordinate of region
• 3 digits system within region (known regions have 533 to 553 systems, so not all values are used)
• 1 digit planet within system (0 is not used, largest known system is 6 planets and 2 moons)

So, putting that all together, there are about 600 trillion systems. If eight bodies per system is typical (in practice this is a high estimate), that's about 5 quadrillion, still far from the 18 quintillion being discussed here.

In practice it doesn't matter, because if a new system were discovered every second, it would take 74,000 years to reach all of the systems in one galaxy.

Update:

As of the "1.5 NEXT" update, there is 4-player multiplayer. Players can see each others' buildings/ships, interact and trade with each other, and fight together. Quests are still separate.

If you're here to see the list of all the ways the players were lied to about multiplayer, see this answer's history.