Why can't I add an existing field to the user accounts on Drupal 8? On Drupal 7, you can add fields that have been added to nodes to user accounts.  At admin/config/people/accounts/fields, scroll down to Add related fields and simply select the field, easy.

But in Drupal 8, at /admin/config/people/accounts/fields, I see no option to add an existing field, just Add Field, which adds a new field.

Has this option been removed from Drupal 8, and if so, why?
Yes this option has been removed, a field cannot be shared across entity types.

You can, however, add the same field again, I.e. use the same name. Whilst they will share a name, they will be separate fields.

The reason for this is storage. In drupal 7, field storage was separate to entity storage, which meant a mismatch. In drupal 8, entity storage also handles field storage. This means if for example mongodb wants to store an entity and it's fields as a single blob/document - it can. On drupal 7 this would have required two blobs, one for the entity and one for the fields.