As shown by the images, they do not have the same simulation region. FDE used metal BCs so the mode you are seeing might not be the actual propagating mode. Instead it might be the box mode formed by the metal BCs.

The modes are very tight in terms of effective index (very similar to the whisper gallery modes). therefore you will need to make sure this is the mode to be compared.

You are right that ” the eigen mode solver solves Maxwells equations in the respective co-ordinate system” however it is not based on the symmetry of the structure, except that you specify the symmetry BCs.

One coordinate system is a projection/transformation of another, so the mode profile may not be necessarily the same.

Please make sure you use exactly the same frequency.