The basic idea to achieve this would be to add some small delay to the click operation, and cancel that if a dblclick is received. This does lead to reduced responsiveness of your UI, as your application is now having to explicitly distinguish between single and double clicks in a kind of "wait and see" approach. This can also be unreliable as the user may set any length of time as their double-click threshold in their device's Accessibility settings.

However, if these issues are deemed not to be enough of a concern, the "best of the worst" way to do this would be to manually implement double-click checks.

export const createTag = (name) => {
  const tag = document.createElement('span');
  tag.innerHTML = name;

  const clickHander = () => {
    search.value = tag.innerHTML;
    const keyboardEvent = new KeyboardEvent('keypress', {
      code: 'Enter',
      key: 'Enter',
      charCode: 13,
      keyCode: 13,
      view: window,
      bubbles: true,
    });
    storeTags.splice(storeTags.indexOf(name), 1);
    storeTags.unshift(name);
    search.dispatchEvent(keyboardEvent);
  };

  const dblClickHandler = () => {
    tags.removeChild(tag);
    storeTags.splice(storeTags.indexOf(name), 1);
  };

  let clickTimer;
  tag.addEventListener('click', () => {
    if( clickTimer) {
      clearTimeout(clickTimer);
      dblClickHandler();
      clickTimer = null;
    }
    else {
      clickTimer = setTimeout(() => {
        clickHandler();
        clickTimer = null;
      }, 500);
    }
  });

  return tag;
};

Adjust the timer as you feel appropriate. Smaller numbers will lead to more responsive clicks, but will make double-clicking harder for people with reduced mobility.

I would really strongly recommend using a different UI here. Overloading clicks in this way is a really bad idea in general.