Why is gymnastics and drama done in bare feet?

Your feet will have better traction than socks.  And shoes would limit the flexibility of the foot, and you need that flexibility in gymnastics.

For gymnastics, bare feet are safer, less likely to slip and generally easier.

For drama, there's no need for bare feet at all.  Most acting students wear suitable footwear, depending on the type of work they're doing.

You need your feet to grip mats/bars/etc. We did wear a kind of gym slippers when I did gymnastics, but I wasn't very advanced.

Your feet will have better traction than socks.  And shoes would limit the flexibility of the foot, and you need that flexibility in gymnastics.

You can always wear the grippy socks for indoor athletics. They also help to avoid verrucas.