Plumbing – 1/4″ Reverse Osmosis Filter inlet line to undersink cold water line

I don't see any problem with installing it horizontally on the run between the two sinks. You'll probably need a new slip nut (the part that goes on top of the trap) but otherwise it can probably connect into the existing union that's there now. Just keep the piece that goes to the RO on the top (facing up).

Where does the hose in that image currently go? Is that an existing dishwasher inlet or does it go somewhere else like a laundry closet or to a sprayer/sink?

Also, we need to know what type of fitting the dishwasher inlet will be. Most that I know of are threaded flex hose.

The part you show does not look like what you want, if you just want to screw it into your existing fitting. This part designed to be inserted into the middle of an ordinary copper pipe using compression fittings that join to an unthreaded copper or nylon pipe end, providing a ball-valve connection in the middle of that pipe suitable for a washing machine. You CAN use this, especially since you have a shutoff valve (which appears to have been installed in much the same way), but you must permanently alter your plumbing by cutting some part of the pipe in half to put this fitting in.

What you probably want is a male-male-female NPT equal tee or Y fitting, allowing you to screw in a second hose exactly like the one you have now. Something like this (from the same site) may fit the bill:

If you can find it in nickel-steel, it'll last a little longer, but this plastic fitting should be perfectly fine. Simple, near tool-less (just a pair of slip-joint pliers or a small strap wrench to tighten down the hoses), and undoable.