Is "Center Midfielder (support ) & Box to Box Midfielder is the same role ?

Box to Box midfielder helps defend and striker so do center midfielder ( support )

No.

Box-to-box midfielder literally runs from box to box (good pace and stamina needed for him!). CM support mostly sticks to central pitch area and supports both defence and attack from his midfield position, therefore needs less stamina/pace than box-to-box midfielder. Physical attributes are also key for choosing your CM role.

Central midfielder (support) is somewhere between CM(a) and CM(d), but box-to-box is more than that- he's both a CM(a) and a CM(d).