1. Let each identical cell have

   • emf $= E$
   • internal resistance $= r$

   External resistance is given as $R = 2\,\Omega$

2. Current when cells are in series

   For two identical cells in series:

   • equivalent emf $= 2E$
   • equivalent internal resistance $= 2r$

   So current through external resistance is $I_s = \frac{2E}{R+2r}$

3. Current when cells are in parallel

   For two identical cells in parallel:

   • equivalent emf $= E$
   • equivalent internal resistance $= \frac{r}{2}$

   So current through external resistance is $I_p = \frac{E}{R+\frac{r}{2}}$

4. Given both currents are same

   $I_s = I_p$

   Therefore, $\frac{2E}{R+2r} = \frac{E}{R+\frac{r}{2}}$

5. Cancel $E$ and solve

   $\frac{2}{R+2r} = \frac{1}{R+\frac{r}{2}}$

   Cross-multiplying, $2\left(R+\frac{r}{2}\right)=R+2r$

   $2R + r = R + 2r$

   $R = r$

6. Substitute $R=2\,\Omega$

   $r = 2\,\Omega$

7. Check options

   • A: $2\,\Omega$ ✅
   • B: $4\,\Omega$ ❌
   • C: $6\,\Omega$ ❌
   • D: $8\,\Omega$ ❌

Hence, the internal resistance of each cell is $\boxed{2\,\Omega}$