(Backlight) LED Driver maximum number of LEDs
Isn't it irrelevant how many LEDs you connect as long as the voltage
  is correct and the driver can supply or sink enough current?


Only the string max voltage and current are relevant not the max number of LEDs.
I don't know why they specify the output in terms of the number of LEDs.  




  Why do some ICs have multiple sink channels for separate strings when
  you can connect them together to increase the total current?  


Flexibility to increase target market.   




  I have 2 backlight strings of 120mA each


And about 24-30V per channel.  

Check out the TI LP8862,  Allegro A8502