This is true only for Teensy 3.6 when running faster than 120 MHz. Higher clock speeds on Teensy 3.6 use a special "HSRUN" mode. Internal EEPROM writing is not possible while in that special mode, so writing to EEPROM temporarily slows the CPU clock. The peripheral clock remains the same, so most timers and peripherals remain at the same speed. But Serial1 and Serial2 on Teensy 3.6 are clocked from the faster CPU clock, so they are affected.
Again, only Teensy 3.6 has HSRUN mode. None of the other Teensy 3.x boards have it. No Teensy 4.x use this special mode. It's unique to Teensy 3.6.