Nah, I'm pretty sure Sony has no legal ownership on how they label their shoulder buttons. Otherwise I can't imagine 8BitDo would be able to get away with labelling their shoulder buttons as L, R, L2, and R2 on all their controllers (except the specifically Xbox-branded ones which use the Xbox LB, RB, LT, and RT labelling instead).L1/R1/L2/R2 makes the most sense, but Sony's probably got that locked down legally. I can at least follow bumper vs. trigger, but ZL/ZR and L/R somehow just makes my brain shit itself. Probably all the years spent with the SNES/N64, associating the long, narrow shoulder buttons with just L and R in a Nintendo context.
It's probably each company just chooses to do their own 'thing' for how they label and refer to the shoulder buttons rather than them actually having any ownership of the specific labels they use.