You are right, it has branch cut discontinuities. But you can correct for them:

solN = Block[{g = 9.8, L = 1}, 
   NDSolveValue[{\[Theta]''[t] + g/L*Sin[\[Theta][t]] == 
      0, \[Theta][0] == Pi/2, \[Theta]'[0] == 0}, \[Theta], {t, 0, 
     10}]];
sol = DSolveValue[{\[Theta]''[t] + g/L*Sin[\[Theta][t]] == 
     0, \[Theta][0] == Pi/2, \[Theta]'[0] == 0}, \[Theta], t];
solDiscontinuities = 
  FunctionDiscontinuities[
   JacobiAmplitude[
    1/2 (-((Sqrt[2] Sqrt[g] t)/Sqrt[L]) + 2 EllipticF[\[Pi]/4, 2]), 
    2], t, Reals];
{discnt2, discnt1} = 
  t /. Solve[solDiscontinuities, t, Reals] /. {C[1] -> 0, g -> 9.8, 
    L -> 1};
solCorrected[t_] := sol[Mod[t, discnt2 - discnt1, discnt1]]
Block[{g = 9.8, L = 1}, {Plot[solCorrected[t], {t, 0, 10}],
  Plot[solN[t], {t, 0, 10}]}]

Look up DSolve:

Clear[g];
DSolve[{\[Theta]''[t] + g/L*Sin[\[Theta][t]] == 0,
  \[Theta][0] == Pi/2, \[Theta]'[0] == 0},
 \[Theta][t], t]