Starting with my first console, the Atari 2600, the controller had one button and a joystick. Designators were not needed. The console itself had start and select buttons to, as their names imply, select a mode and start the game. While most controllers still have these buttons, they seem to be delegated to pausing and certain menu functions.
When the NES debuted, its 2-button system required differentiation. Two popular methods would be alpha or numeric. They chose alpha. It would make sense to begin with "A" and progress alphabetically until all buttons are named, in this case proceeding all the way to "B".
The Sega Genesis arrived on store shelves with "A" "B" and "C" buttons. When the SNES followed with a 6-button system, the choice to use "X" and "Y" rather than "C" and "D" was most likely to differentiate the product from its rival. Labeling the shoulder buttons "L" and "R" obviously denote the physical location of the button, but also imply a difference in function, subtly suggesting that they are distinct from the 4 primary buttons.
Sony's geometric shapes were likely an effort towards product branding, seeking to create an interface that was unique in appearance. Given the fact that the button layout was nigh identical to the SNES controller (barring the addition of two additional shoulder buttons), this product differentiation was important.
Skipping a few consoles ahead to the X-box, the use of the "ABXY" format was most likely to take advantage of it's familiarity with gamers. There wasn't much else they could have done to distinguish themselves aside from using a numeric system, but the entrenched popularity of the alpha system would have likely made that a poor decision.
