Kuba's MoreCalculus package - an example

As noted by Kuba, there is a typo in the PDE. Then, to obtain 2nd PDE in the question the needed rule is

\begin{equation} \begin{aligned} \color{red}{\eta} &= \rho \cos w \,\\ \color{red}{\sigma} &= \rho \sin w \,\\ \end{aligned} \end{equation}

Finally, your transformation rule is improper. Try the following:

Assuming[{ρ > 0, -Pi < w < Pi}, 
 DChange[1/σ^2 D[σ^2 D[V[σ, η], σ], σ] + D[V[σ, η], {η, 2}] == 0, 
   "Cartesian" -> "Polar", {η, σ}, {ρ, w}, {V[σ, η]}]]

Or the following:

Assuming[{ρ > 0, -π < w < π}, 
 DChange[D[σ^2 D[V[σ, η], σ], σ]/σ^2 + D[V[σ, η], {η, 2}] == 0, 
         {Sqrt[η^2 + σ^2] == ρ, w == ArcTan[η, σ]}, {η, σ}, {ρ, w}, {V[σ, η]}]]

Thanks and sorry but I won't have time to update the package soon nor I were doing any calculus for couple of years so here are just my quick notes (which do not solve everything):

You forgot about 1/σ^2 in your equation.

The question is what to assume to make:

Solve[{σ == ρ*Cos[w], η == ρ*Sin[w]}, {ρ,  w}]

to return only the second result. I tried ρ > 0 but that makes it stuck.

Regardless,

DChange[
  1/σ^2 D[σ^2*D[V[σ, η], σ], σ] + D[V[σ, η], {η, 2}] == 0,
  {σ==ρ*Cos[w],η==ρ*Sin[w]},
  {σ,η},{ρ,w},{V[σ,η]}
] //
  ReplaceAll[C[1]->0] //
  Refine[#,{ρ>0}]&    // (*Get rid of Sqrt[ρ^2] ?justified*)
  ReplaceAll[ArcTan[-Cos[w],-Sin[w]]->w] // (*maybe can be automatic with smart Assumptions*)
  Map[-#/ρ &] // Expand // (* tidy up*)
  TraditionalForm

Closer but not yet there