Hi Alberto,
The uu and vv from the tutorial are there in order to define the bilinear form needed for the weak formulation of the problem. For an introduction see : Weak formulation - Wikipedia. The (u,v) from your question is the “u” from the General Concept section, while the (uu,vv) from your question is the “v” from the General Concept section.
In the 3D case you can call the 3 unknown components of the displacement vector (u,v,w) and of course you can call them (Alex1, Alex2, Alex3) since it is just a name… In addition, you need also a vector test function that you can call (uu,vv,ww), or any other name like (u1,v1,w1).
Best regards, Alex