Motivation: the permutations of the projective space $P(V)$ can be regarded as the equivalence classes of such functions (two such functions, $f$ and $g$ are regarded to be equivalent if for every 1-dimensional subspace $L$ of $V$, $\{f(x):x\in L\}=\{g(x):x\in L\}$). I would call them projective functions but I'm afraid that this name means something else.
Edit
I definitely mean of all such functions, not only the homogeneous (or more specifically, the linear) ones.
Here is how to construct an arbitrary function $\phi:V\to V$ sending 1D subspaces to other 1D subspaces (which I will just call "lines" for short). Picking any two lines $L_1$ and $L_2$, we know $0$ must be mapped to a point on both lines $\phi(L_1)$ and $\phi(L_2)$, which is just $0$, so $\phi$ fixes $0$. It remains to see how $\phi$ can act on nonzero vectors. We can split this up into two parts: how $\phi$ acts on the set $P=\mathbb{P}V$ of all lines, and how it permutes points within lines.
For every function $\sigma:P\to P$, there are going to be $\phi$s which send lines $L$ to lines $\sigma(L)$. Note the restriction of $\phi$ to a line must be a bijection. Thus, we can decompose the set $\Phi$ of all such $\phi$s into
$$ \Phi = \bigsqcup_{\sigma:P\to P} \prod_{L\in P} \mathrm{Bij}(L,\sigma(L)) $$
where $\mathrm{Bij}(X,Y)$ denotes the set of bijections $X\to Y$ and the lines are considered to not contain $0$. As a monoid, this is a kind of "monoid wreath product" of the monoid of all $\sigma:P\to P$ and the direct product of symmetric groups $S_L$ for all lines $L$ (again, ignoring $0$). This might be easier to understand by looking at the subgroup $G$ of invertible $\phi$s.
First, we can construct a copy of $S_P$ within $G$ that permutes the lines themselves as follows. For each line $L$, choose a bijection $\ell_L:L\to k^{\times}$ (where $k$ is the base field), which induces a bijection between any two lines as $\ell_{L_2}\circ\ell_{L_1}^{-1}:L_1\to L_2$. Then, given any permutation $\pi\in S_P$, we can think of it as a function on $V$ (fixing $0$) by saying $\pi$'s restriction to any line $L$ is $\ell_{\pi(L)}\circ\ell_L^{-1}$. This givecs a copy of $S_P$ within $G$.
The subgroup of $G$ which fixes every line is a direct product $\prod_{L\in P}S_L$. Note this direct product intersects $S_P$ trivially, and that $S_P$ normalizes this direct product. Moreover, if $\phi\in\Phi$ is arbitrary (well, invertible) and induces a permutation $\pi$ of $P$ then $\phi\circ\pi^{-1}$ fixes each line so is an element of the direct product. That means
$$ G= \big(\prod S_L\big)\rtimes S_P \cong S_{k^{\times}}\wr S_P $$
is a wreath product.