Light multiple RGB Led Strips from same 3 RGB PWM outputs
You have few options:


do as you represented in the drawing: it should work, as I expect the transistors to not load too much the input.
add a logic buffer inbetween the arduino and the driver stage
use other pins from the arduino for driving the additional strips and bitbang the PWM, instead of using the HW blocks


Whether the solution you propose (1) will work or not depends entirely on the max current that each PWM pin can output vs the total load from the 3 transistors it will have to drive.