In the original thread Carmack was replying to, Sandy Petersen said Q3 was the only other great game they produced after that: https://xcancel.com/SandyofCthulhu/status/206959226489744192...
Honestly I think Doom is where it came together the best, Quake was technically better (of course) but it was not a better game.
The true successor of ID software is Half-life 1 with its goldsrc engine... but that simply was made by another studio.
HL1 took both the engine and the genre further + continued the modding culture that brough Counter strike and other mods
(Note I know very well that Half life is not an ID software game, it only took the engine that was auper heavily modified / updated- but it my opinion this is the successor)
Quake was better for multiplayer though. Personally I enjoyed Quake 2 the most. Quake arena was designed for multiplayer but you had to practice with the cheating bots so it was kinda boring.