Yes you do. It's created when you register the device to Nintendo's servers. You're just giving it a display name, but the unique ID is still limited to that server-generated numeric ID, not a unique alphanumeric ID that you can choose for yourself.
Okay, I'll be more clear.
The Wii has
1) A system based friendcode
2) Software based friendcodes
The 3DS has
1) A system based friendcode
You are talking about the automated account creation of the 3DS/Wii. While I am talking about manual account creation like the PS3/360/Steam.
For example,
You boot up your new system.
1) Do you have an account? Y/N
2) Would you like to connect to the internet and create an account? Y/N
3) Your ID is ~random number
4)Create your Mii
5)Assign it's name (your display name)
6)Please enter the following information
E-mail address
Password
Birthday
Ect ...
7) Would you like to make this the master account? Y/N
Now you have created an account based around a numeric value. It makes the process more streamlined than on the 360/PS3
So whenever you want to log in to your account from your Wii U you would just select your Mii than type your password. When you add friends their Mii will be stored on your system too, so if they ever come over all they need to do is select their Mii and enter their password.
_
Now if you get a new Wii U or use someones you aren't friends with you could search for your Mii using your email address, add it then log-in.