entities – How can I load an entity’s bundle object from a loaded entity?

The bundle field is a reference field and you can load the bundle object via the field property entity:

$node_type = $node->type->entity;

More generic for any content entity:

if ($entity->getEntityType()->hasKey('bundle')) {
  $bundle = $entity->get($entity->getEntityType()->getKey('bundle'))->entity;
}