How do I add secondary tabs to the user (profile) edit tab?
Tabs can be added to the user edit page as follows:

1) Define the route(s) for your tab(s) in [MODULE].routing.yml as normal:

[MODULE].something.user_defaults:
  path: '/user/{user}/edit/something'
  defaults:
    _controller: '\Drupal\[MODULE]\Controller\SomeController::someSettingsTab'
    _title: 'Some Settings'
  requirements:
    _permission: 'some permission'
    user: \d+
  options:
    parameters:
      user:
        type: entity:user


2) In [MODULE].links.task.yml, add your tab(s) as you would primary tabs, except rather than using base_route, use parent_id, and set it to (somewhat unintuitively) the user edit form tab key, which is defined in user.links.task.yml as entity.user.edit_form (same as the route name):

[MOUDLE].something.user_defaults:
  route_name: [MODULE].something.user_defaults
  parent_id: entity.user.edit_form
  title: Some Title
  weight: 10


3) You also need to define a tab for the user to switch back to, as tabs are only rendered if there are more than two tabs. In this case, we want to add a tab for the user edit form, using the user edit form route, again using the user edit form tab key:

[MODULE].entity.user.edit_form:
  route_name: entity.user.edit_form
  parent_id: entity.user.edit_form
  title: Profile