This can be achieved with some creativity:
SET @sql = CONCAT('INSERT INTO <table> SELECT null,
', (SELECT GROUP_CONCAT(COLUMN_NAME)
FROM information_schema.columns
WHERE table_schema = '<database>'
AND table_name = '<table>'
AND column_name NOT IN ('id')), '
from <table> WHERE id = <id>');
PREPARE stmt1 FROM @sql;
EXECUTE stmt1;
This will result in the new row getting an auto incremented id instead of the id from the selected row.