Just to add onto this because I can't seem to find any talk about this anywhere else yet: Pillow toss's on-hit cooldown effect is seemingly intended to affect the cooldown on Lil Helpers, because it lights up like his other two abilities when triggered (and as mentioned by the OP, cooldown reduction items affect the 15.1s cooldown that begins after following a teammate or minion for their respective 8 or 50s durations).
Unsure if this is a glitch or not, but if it isn't then I feel like more clarity should be offered (e.g. UI effect no longer appears on the Lil Helpers' ability icon, and Pillow Tosses' tooltip specifies reducing Tag Along / Naptime exclusively).
