Where would I put the code in my Vue application?
You would put the code for the additional signup fields in the custom login page of the Classic ULP (Universal Login Page), which is to be found under the Auth0 Dashboard > Universal Login > Login tab > check "Customize login page".
So, the code doesn’t go in your Vue app but into the Auth0 hosted ULP.
(Note that the additional signup fields only work with the Classic ULP, not the New ULP.)
Is Lock the same as Universal Login?
It’s not exactly the same, it’s more that: the Classic ULP uses the Lock.js library internally, as you can see from the screenshot above.
The Lock.js is a library which is used by the Auth0 hosted ULP, but it can also be embedded by client applications (which we usually don’t recommend though). So, I fully understand that it can get a bit confusing here.
(The New ULP however does not use Lock.js or any Javascript at all, which is why the additional fields only work with the Classic ULP)