You CAN,but it's not necessary in order to have a strong core,and if you were to train it every day,in my opinion,you would certainly be doing more harm than good if you don't mix it up as much as possible-isometric (i.e. plank) twisting,targetting lower or upper abs or obliques as appropriate on different days,with different exercises,etc. instead of a total beasting every day.
doing say,sit-ups and crunches every day would seem a bad idea unless it was very specific,i.e. boxing/martial arts or correcting a severely lordotic spine.
I do train my abs everyday, but I alternate on heavy standing ab work on my training days, and when I do recovery/extra workouts, I tend to do bw stuff like Hanging Leg Raises to build up work capacity and/or recovery.