SELECT  LPAD(' ', 2  * LEVEL) || ID_ANIMAL
  FROM  ANIMALS
  START WITH    ID_MOTHER IS NULL
             OR
                ID_FATHER IS NULL
  CONNECT BY    PRIOR ID_ANIMAL = ID_FATHER
             OR
                PRIOR ID_ANIMAL = ID_MOTHER
/

 SELECT  -(LEVEL -  1 ) GENERATION,
         ID_ANIMAL
   FROM  ANIMALS
   START WITH ID_ANIMAL = some_id
   CONNECT BY ID_ANIMAL = PRIOR ID_FATHER
                 OR ID_ANIMAL = ID_MOTHER
UNION
 SELECT  LEVEL -  1  GENERATION,
         ID_ANIMAL
   FROM  ANIMALS
   START WITH ID_ANIMAL = some_id
   CONNECT BY PRIOR ID_ANIMAL = ID_FATHER
                 OR ID_ANIMAL = ID_MOTHER
/

SELECT  GENERATION,
        NAME
  FROM  (
          SELECT  DECODE(
                         LEVEL,
                          1 ,'Self',
                          2 ,'Parent',
                          3 ,'Grandparent',
                         LPAD('Great ',(LEVEL - 3 ) *  5 ,'Great ') || 'Grandparent'
                        ) GENERATION,
                  ID_ANIMAL
            FROM  ANIMALS
            START WITH ID_ANIMAL = some_id
            CONNECT BY    ID_ANIMAL = PRIOR ID_FATHER
                       OR
                          ID_ANIMAL = PRIOR ID_MOTHER
         UNION
          SELECT  DECODE(
                         LEVEL,
                          1 ,'Self',
                          2 ,'Child',
                          3 ,'Grandchild',
                         LPAD('Great ',(LEVEL - 3 ) *  5 ,'Great ') || 'Grandchild'
                        ) GENERATION,
                  ID_ANIMAL
            FROM  ANIMALS
            START WITH ID_ANIMAL = some_id
            CONNECT BY    PRIOR ID_ANIMAL = ID_FATHER
                       OR
                          PRIOR ID_ANIMAL = ID_MOTHER
        )
ORDER BY LEVEL
/

To get all relatives of a specific animal, you would have to combine two queries:
ascendants (parents, grandparents, etc.) and descendants (children, grandchildren, etc.): 

 SELECT  -(LEVEL -  1 ) GENERATION,
         ID_ANIMAL
   FROM  ANIMALS
   START WITH ID_ANIMAL = some_id
   CONNECT BY    ID_ANIMAL = PRIOR ID_FATHER
              OR
                 ID_ANIMAL = PRIOR ID_MOTHER
UNION
 SELECT  LEVEL -  1  GENERATION,
         ID_ANIMAL
   FROM  ANIMALS
   START WITH ID_ANIMAL = some_id
   CONNECT BY    PRIOR ID_ANIMAL = ID_FATHER
              OR 
                 PRIOR ID_ANIMAL = ID_MOTHER
/

SELECT  GENERATION,
        ID
  FROM  (
          SELECT  -(LEVEL -  1 ) LVL,
                  DECODE(
                         LEVEL,
                          1 ,'Self',
                          2 ,'Parent',
                          3 ,'Grandparent',
                         LPAD('Great ',(LEVEL - 3 ) *  6 ,'Great ') || 'Grandparent'
                        ) GENERATION,
                  ID_ANIMAL
            FROM  ANIMAL
            START WITH ID_ANIMAL = some-id
            CONNECT BY    ID_ANIMAL = PRIOR ID_FATHER
                       OR
                          ID_ANIMAL = PRIOR ID_MOTHER
         UNION
          SELECT  (LEVEL -  1 ) LVL,
                  DECODE(
                         LEVEL,
                          1 ,'Self',
                          2 ,'Child',
                          3 ,'Grandchild',
                         LPAD('Great ',(LEVEL - 3 ) *  6 ,'Great ') || 'Grandchild'
                        ) GENERATION,
                  ID_ANIMAL
            FROM  HIERARCHYX
            START WITH ID_ANIMAL = some-id
            CONNECT BY    PRIOR ID_ANIMAL = ID_FATHER
                       OR
                          PRIOR ID_ANIMAL = ID_MOTHER
        )
ORDER BY LVL
/