Its a myth about kicking a dog-tag into the mouth, try crouching down holding onto a thin sliver of metal and kicking it at the same time!!
Its rare for anybody to die with their teeth tightly clenched together the tags are only about 3mm in thickness so they slip in easy, and if push came to shove you could pry the teeth open with the bayonet, which is certainly more dignified than kicking your dead comrade in the face to insert the dog-tag!!!
The reason one tag is left with the body is so that if placed in a temporary battlefield grave, later when the body is recovered by the Graves Registration Unit they can identify it and record it.
Many soldiers also have a spare pair laced into their boots, one on each boot lace, in case the head gets blown off and the ones round the neck get lost.