We have constant current LED strips too (like this one ). We can leave the power LEDs out and use only constant voltage LED strips so we can adapt to both situations, whichever is the better solution.
That strip seems to be a mix of common CL resistors and a tiny linear regulator (I don't see any inductors).
I would still call this a constant voltage strip (with a little bit of regulation).
I don't see any voltage window limits in the specs where the regulation would still work or fry the linear chip.
This strip should be PWM-ed low-side with a locic level mosfet.
There are many low voltage CC LED drivers available for the Arduino,
but they need to be matched to the LED you're driving.
Give us a link to the 'other' LEDs you want to control.
Leo..