When serialising the datbase connection object we remove the connection, schema and driverClasses objects before hand and reinitialise them when the string is unserialised. However, when recreating the PDO connection class we don't set ATTR_STATEMENT_CLASS again so any statements created use the default PHP PDO class which doesn't have all of the methods Drupal expects and so can lead to issues.
As a quick fix I've added in some code to set the attribute again but it might be better to look into calling out to the constructor again to avoid duplicating code.