'Stream-ripping' (which is what you're contemplating) is only legal where the copyright holder has granted permission for it. That applies to Youtube, DailyMotion, Facebook and anywhere else videos are posted online.
So, for example, it's illegal to save videos (or just the audio content of them) of your favourite band from Youtube.
However (although it's still technically illegal without consent) it's unlikely that someone would object to you saving the video that she's posted online of her kittens climbing up her curtains.