Maybe this helps:

system = {x'[t] == 1 - y[t], y'[t] == x[t]^2 - y[t]^2};
ibc = {y[0] == 1, y'[0] == 2};
sol = NDSolve[{system, ibc}, {x[t], y[t]}, {t, 0, 5}];
Plot[Evaluate[{x[t], y[t]} /. sol], {t, 0, 5},PlotLegends -> {"x[t]", "y[t]"}, PlotRange -> All]

Because yours initial condition y'[0]=2 is very unusual(should not to be).In this case initial condition should be: {x[t0]=x0,y[t0]=y0}. Put:

ibc = {y[0] == 1, x[0] == 2};

and works fine.

You can use a ParametricNDSolve to plot some other curves.

system = {x'[t] == 1 - y[t], y'[t] == x[t]^2 - y[t]^2};
sol = ParametricNDSolve[{system, y[0] == a, x[0] == b}, {x, y}, {t, 0, 5}, {a, b}];
Plot[Evaluate[Table[x[a, b][t] /. sol, {a, 1, 5, 1}, {b, 1, 5, 1}]], {t, 0, 5}, PlotLegends -> {"x[t]"}, PlotRange -> All]

Plot[Evaluate[Table[y[a, b][t] /. sol, {a, 1, 5, 1}, {b, 1, 5, 1}]], {t, 0, 5}, PlotLegends -> {"y[t]"}, PlotRange -> All]

For more help of ParametricNDSolve