How can I display the field in the tab that was active on save?
I think it is being called multiple times. To avoid that try Change your JS as below

$(context).find('.single-link').click(function () {
  $(context).find('.option').prop('checked', true);
});
$(context).find('.multi-link').click(function () {
  $(context).find('.option').prop('checked', false);
});


OR

change the on click() event to hasClass() the will be classes added to the active tab