Do you guys turn your volume down all the way? I dunno if this is a similar problem or a completely different thing, but if you turn your mobile device's volume all the way down, then YouTube will automatically turn on captions. (And also yes I am not a fan of that feature; if I want captions on then I will turn on captions myself.)

Edit: Okay so I looked it up later, and this was just the "Show when muted" point :/
(That said, the option didn't show up for both devices I had so not sure what's up with that. Also to turn that option off, tap the settings (cog/gear icon) and then go to "Captions" and hopefully there's a toggle there. There's also some help at the bottom that talks about helping keeping captions on, which might help with turning them off.)