As one of the possibly solutions we can decrease accuracy and use explicit method as follows

    ClearAll["Global`*"]
    (*Parameters*)
    xRef = 10^(-8);
    rhoRef = 0.1*nA;
    j = 0.5; (*normalized conc/(m^2s)*)
    rhoFStarVal = 120;(*normalized fixed charge concentration*)
    rhoBStarVal = 20;(*normalized bulk concentration*)
    phiBC = -0.25;(*in thermal voltage unit*)
    kB = 1.380649*10^-23;
    absTemp = 298.15;
    beta = 1/(kB*absTemp);
    eps0 = 8.8541878128*10^(-12);
    eps = 78.6;
    e = 1.60217663*10^(-19);
    nA = 6.02214076*10^(23);
    dfsvt = 1.33*10^(-9);
    xL = 0;(*in xRef*)
    xMemL = 0.75; (*in xRef*)
    xMemR = 1.25;(*in xRef*)
    xR = 2;(*in xRef*)
    stepFunc[x_] = 0.5*(Tanh[100*x] + 1);
    filterFunc[x_] = stepFunc[x - xMemL]*stepFunc[-(x - xMemR)];
    rhoFStar[x_] = rhoFStarVal*filterFunc[x];
    zf = -1;
    z1 = 1;(*Counter ion valence*)
    z2 = -1;(*Co ion valence*)
    deltaU = -1/beta;
    vW = 30*10^(-30);
    debye2 = (kB*absTemp*eps*eps0)/(e^2*rhoRef);
    
    (*Equations*)
    elctrcTrnsprtV2 = 
     debye2/xRef^2*
       phi''[x] == -(zf*rhoFStar[x] + z1*rho1fStar[x] + z2*rho2fStar[x])
    ctionTrnsprt = 
     rho1fStar'[x] == -z1*rho1fStar[x]*phi'[x] - (j*xRef)/dfsvt
    coionTrnsprt = rho2fStar'[x] == -z2*rho2fStar[x]*phi'[x]
    
    (*NDSolve*)
    
    totalSolV2 = 
     NDSolve[{ctionTrnsprt, coionTrnsprt, elctrcTrnsprtV2, phi[xL] == 0, 
       phi[xR] == phiBC, rho1fStar[xL] == rhoBStarVal, 
       rho2fStar[xL] == rhoBStarVal}, {phi, rho1fStar, rho2fStar}, {x, xL,
        xR}, AccuracyGoal -> 5, PrecisionGoal -> 5, 
      Method -> "ExplicitEuler"]
    
    
    (*Define the concentration function concentration[r,t]*)
    phiSol = totalSolV2[[1]][[1]][[2]];
    rho1fStarSol = totalSolV2[[1]][[2]][[2]];
    rho2fStarSol = totalSolV2[[1]][[3]][[2]];
    
    (*Solution Plotting*)
    
    {Plot[phiSol[x], {x, xL, xR}, PlotRange -> All, 
      PlotLabel -> "phi (KbT/e), steady"],
     Plot[rho1fStarSol[x], {x, xL, xR}, PlotRange -> {0, All}, 
      PlotLabel -> "Free Counter Ion (0.1 mM/L), steady"],
     Plot[rho2fStarSol[x], {x, xL, xR}, PlotRange -> Full, 
      PlotLabel -> "Free Co Ion (0.1 mM/L), steady"]} 

 [![Figure 1][1]][1]

Second method is the Shooting method

    sol2 = 
     NDSolve[{ctionTrnsprt, coionTrnsprt, elctrcTrnsprtV2, phi[xL] == 0, 
       phi[xR] == phiBC, rho1fStar[xL] == rhoBStarVal, 
       rho2fStar[xL] == rhoBStarVal}, {phi, rho1fStar, rho2fStar}, {x, xL,
        xR}, AccuracyGoal -> 5, PrecisionGoal -> 5, 
      Method -> {"Shooting", 
        "StartingInitialConditions" -> {phi[xL] == 0, phi'[xL] == -.7, 
          rho1fStar[xL] == rhoBStarVal, rho2fStar[xL] == rhoBStarVal}}]

Visualization

    {Plot[phi[x] /. sol2[[1]], {x, xL, xR}, PlotRange -> All, 
      PlotLabel -> "phi (KbT/e), steady"],
     Plot[rho1fStar[x] /. sol2[[1]], {x, xL, xR}, PlotRange -> {0, All}, 
      PlotLabel -> "Free Counter Ion (0.1 mM/L), steady"],
     Plot[rho2fStar[x] /. sol2[[1]], {x, xL, xR}, PlotRange -> Full, 
      PlotLabel -> "Free Co Ion (0.1 mM/L), steady"]}
[![Figure 2][2]][2]


  [1]: https://i.sstatic.net/kmSFkdb8.png
  [2]: https://i.sstatic.net/7UsZgTeK.png