Why $f(x,y)=y^2$ instead of $f(y)=y^2$? If I have a function $f(x,y)=y^2$, is it equivalent with $f(y)=y^2$?
Are there any differences? 
Is $f(x,y)=y^2$ a multivariable function despite no $x$?
 A: $f(x,y) = y^2$ is a function of two variables that happens to depend on just one of them.  It is not the same as $f(y) = y^2$, which is a function of one variable, but for some purposes it can be regarded as equivalent.
A: $f(x,y)=y^2$ is a function from $\mathbb{R}^2$ to $\mathbb{R}$, while $f(y)=y^2$ is function from $\mathbb{R}$ to $\mathbb{R}$, so no, they are not the same function. 
See this recent answer for what must be true in order for two functions to be equivalent.
A: Let
$$\begin{array}{rcl} f \,\, : & \mathbb R^2 &\to \mathbb R\\ & (x,y) &\mapsto y^2\end{array}$$
and
$$\begin{array}{rcl} g \,\, : & \mathbb R &\to \mathbb R\\ & y &\mapsto y^2\end{array}$$
Thus, $g$ is the restriction of $f$ to the $y$-axis $\{(0,\gamma) \mid \gamma \in \mathbb R\}$ and $f$ is the extension of $g$ to $\mathbb R^2$. In terms of the graphs of $f$ and $g$, the graph of $g$ is the projection of the graph of $f$ onto the $yz$-plane and the graph of $f$ is the lifting of the graph of $g$ along an axis parallel to the $x$-axis.
