There's a factor of
$(-1)^{k/3}$ for
$k=0,1,2$ in the exponents. Mathematica is finding complex solutions to the ODE as well as real ones. I typed
FullSimplify[DSolve[{D[u[x,t],t]+t^2 D[u[x,t],x]==u[x,t],u[x,0]==Exp[2x]},u,{x,t}]]
TeXForm[u[x,t]/.%]
to obtain the
$\TeX$ code which produces
$$\left\{e^{-\frac{2 t^3}{3}+\sqrt[3]{t^3}+2 x},e^{-\frac{2 t^3}{3}-\sqrt[3]{-1} \sqrt[3]{t^3}+2 x},e^{-\frac{2t^3}{3}+(-1)^{2/3} \sqrt[3]{t^3}+2 x}\right\}$$
Maybe this is nicer to parse visually. It's also not simplifying
$\sqrt[3]{t^3}$ because
$\sqrt[3]\cdot$ is a multivalued function. There are ways to specify that
$x,t$ and
$u$ should be real, but I'm struggling to do that with the methods I know.