= Solution
The three coordinate axes are the three <irreducible components> of
$$
V=V(xy,xz,yz)\subseteq\mathbb A_k^3.
$$
Their common intersection is the origin. Likewise, if $\ell_1,\ell_2,\ell_3$ are distinct linear forms defining the three lines in $\mathbb A_k^2$, then
$$
W=V(\ell_1\ell_2\ell_3),
$$
and its three irreducible components have only the origin in common. Any <isomorphism of algebraic varieties> would permute irreducible components and therefore map the origin of $V$ to the origin of $W$.
Every defining polynomial $xy,xz,yz$ has zero differential at the origin, so
$$
T_0V=k^3
\quad\text{and}\quad
\dim T_0V=3.
$$
The defining polynomial of $W$ is homogeneous of degree three, so its differential also vanishes at the origin, giving
$$
T_0W=k^2
\quad\text{and}\quad
\dim T_0W=2.
$$
An isomorphism induces an isomorphism of <Zariski tangent space>[Zariski tangent spaces], which is impossible here. Hence $V$ and $W$ are not isomorphic. This is the <tangent-space obstruction between two unions of three lines>.
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