MySQL 8.0

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CREATE TABLE table_name (value VARCHAR(200));
INSERT INTO table_name (value)
SELECT 'Object.Name=ASDD||Product.Name=DSA' FROM DUAL UNION ALL
SELECT 'Product.Name=QWE||Object.Name=WSXS' FROM DUAL UNION ALL
SELECT 'Storage.Name=12345||Object.Name=WERR||Product.Name=QAZ' FROM DUAL;
SELECT value,
       CASE
       WHEN start_pos = 0
       THEN NULL
       ELSE SUBSTRING(
              CONCAT('||', value, '||'),
              start_pos + LENGTH('||Object.Name='),
              end_pos - start_pos - LENGTH('||Object.Name=')
            )
       END AS object_name
FROM   (
  SELECT value,
         LOCATE(
           '||Object.Name=',
           CONCAT('||', value, '||')
         ) AS start_pos,
         LOCATE(
           '||',
           CONCAT('||', value, '||'),
           LOCATE('||Object.Name=', CONCAT('||', value, '||'))
             + LENGTH('||Object.Name=')
         ) AS end_pos
  FROM   table_name
) t


value
object_name


Object.Name=ASDD||Product.Name=DSA
ASDD

Product.Name=QWE||Object.Name=WSXS
WSXS

Storage.Name=12345||Object.Name=WERR||Product.Name=QAZ
WERR

value	object_name
Object.Name=ASDD\|\|Product.Name=DSA	ASDD
Product.Name=QWE\|\|Object.Name=WSXS	WSXS
Storage.Name=12345\|\|Object.Name=WERR\|\|Product.Name=QAZ	WERR