PostgreSQL Recursive Descendant Depth

The query you have is basically correct. The only mistake is in the second (recursive) part of the CTE where you have:

INNER JOIN descendants d ON d.parent_id = o.object_id

It should be the other way around:

INNER JOIN descendants d ON d.object_id = o.parent_id 

You want to join the objects with their parents (that have already been found).

So the query that calculates depth can be written (nothing else changed, only formatting):

-- calculate generation / depth, no updates
WITH RECURSIVE descendants
  (id, customer_id, object_id, parent_id, ancestor_id, depth) AS
 AS ( SELECT id, customer_id, object_id, parent_id, ancestor_id, 0
      FROM objects
      WHERE object_id = parent_id

      UNION ALL

      SELECT o.id, o.customer_id, o.object_id, o.parent_id, o.ancestor_id, d.depth + 1
      FROM objects o
      INNER JOIN descendants d ON  d.customer_id = o.customer_id
                               AND d.object_id = o.parent_id  
      WHERE d.id <> o.id
    ) 
SELECT * 
FROM descendants d
ORDER BY id ;

For the update, you simply replace the last SELECT, with the UPDATE, joining the result of the cte, back to the table:

-- update nodes
WITH RECURSIVE descendants
    -- nothing changes here except
    -- ancestor_id and parent_id 
    -- which can be omitted form the select lists
    ) 
UPDATE objects o 
SET generation = d.depth 
FROM descendants d
WHERE o.id = d.id 
  AND o.generation = -1 ;          -- skip unnecessary updates

Tested on SQLfiddle

Additional comments:

• the ancestor_id and the parent_id are not needed to be in the select list (ancestor is obvious, parent a bit tricky to figure out why), so you can keep them in the SELECT query if you want but you can safely remove them from the UPDATE.
• the (customer_id, object_id) seems like a candidate for a UNIQUE constraint. If your data comply with this, add such a constraint. The joins performed in the recursive CTE would not make sense if it wasn't unique (a node could have 2 parents otherwise).
• if you add that constraint, the (customer_id, parent_id) would be a candidate for a FOREIGN KEY constraint that REFERENCES the (unique) (customer_id, object_id). You most probably do not want to add that FK constraint though, since by your description, you are adding new rows and some rows can reference others that haven't been yet added.
• There are certainly problems with the efficiency of the query, if it's going to be performed in a big table. Not in the first run, as almost the whole table will be updated anyway. But the second time, you'll want only new rows (and those that were not touched by the 1st run) to be considered for update. The CTE as it is will have to build a big result.
  The AND o.generation = -1 in the final update will make sure that the rows that were updated in the 1st run will not be updated again but the CTE is still an expensive part.

The following is an attempt to address these issues: improve the CTE as to consider as few rows as possible and use (customer_id, obejct_id) instead of (id) to identify rows (so id is completely removed from the query. It can be used as the 1st update or a subsequent:

WITH RECURSIVE descendants 
  (customer_id, object_id, depth) 
 AS ( SELECT customer_id, object_id, 0
      FROM objects
      WHERE object_id = parent_id
        AND generation = -1

      UNION ALL

      SELECT o.customer_id, o.object_id, p.generation + 1
      FROM objects o
        JOIN objects p ON  p.customer_id = o.customer_id
                       AND p.object_id = o.parent_id
                       AND p.generation > -1
      WHERE o.generation = -1

      UNION ALL

      SELECT o.customer_id, o.object_id, d.depth + 1
      FROM objects o
      INNER JOIN descendants d ON  o.customer_id = d.customer_id
                               AND o.parent_id = d.object_id
      WHERE o.parent_id <> o.object_id
        AND o.generation = -1
    )
UPDATE objects o 
SET generation = d.depth 
FROM descendants d
WHERE o.customer_id = d.customer_id
  AND o.object_id = d.object_id
  AND o.generation = -1        -- this is not really needed

Note how the CTE has 3 parts. The first two are the stable parts. The 1st part find the root nodes that haven't been updated before and have still generation=-1 so they must be newly added nodes. The 2nd part finds children (with generation=-1) of parent nodes that have previously been updated.
The 3rd, recursive part, finds all the descendants of the first two parts, as before.

Tested on SQLfiddle-2