I was trying to use the shop thing and I do not have a Wii U and I don't want to charge my mom 50 cents without her knowing. Please try to help! I am on my 3DS as a type this by the way.
YouTube for Nintendo 3DS can be installed via the official way, but unfortunately, you will still have to use a Nintendo ID. According to the official website, this is the only requirement besides Internet connectivity. To read more, access the website where this process is explained.
Nintendo website: http://en-americas-support.nintendo.com/app/answers/detail/a_id/2065/~/how-to-download-youtube
