How can I plot the two functions below with same color for each value of k?

Evaluate the complete plot-argument:

Show[Map[Plot[ {vest[#, M], bb[#, M]},{M,0, 1},Evaluated -> True, 
PlotStyle -> RGBColor[#/Max[k],0,1-#/Max[k]]] &, k], PlotRange -> All]

m = 10^-3;
k = {1, 5, 10, 25};
ron = 50m;
dcr = k ron;
vest[k_, M_] := k ron + (2 + (-1 + M) M (5 + M)) ron/(-1+M)^2;
bb[k_, M_]:= (2 ron+k ron);
Plot[
 Evaluate @ Flatten @ {Map[{vest[#,M],bb[#,M]}&]@k}]
 ,{M,0,1}
 ,PlotStyle-> {Green,Green,Red,Red,Blue,Blue,Orange,Orange}
]

Another possibility is to use Style wrappers around the individual functions. This enables you to have both data and appearance specified in the same place:

m = 10^-3;
k = {1, 5, 10, 25};
ron = 50 m;
vest[k_, M_] := k ron + ((2 + (-1 + M) M (5 + M)) ron)/(-1 + M)^2;
bb[k_, M_] := M (2 ron + k ron);
Plot[
 Evaluate[
  Style[
     {vest[#, M], bb[#, M]}, 
     ColorData["Rainbow"][#/Max@k]
    ] & /@ k
  ],
 {M, 0, 1}
 ]

Note the use of Evaluate, similar to the other answers.