Is this just with the PDO driver or with mysqli as well?

the code for Joomla PDO database driver is here https://github.com/joomla/joomla-cms/blob/staging/libraries/joomla/database/driver/pdo.php and here for pdomysql specific https://github.com/joomla/joomla-cms/blob/staging/libraries/joomla/database/driver/pdomysql.php

Maybe you can make a Pull Request for solving the issue,

This happens only if using PDOMysql driver. The imySQL driver works perfectly.

I looked earlier. Joomla manually appends the executed SQL query to the error message that the underlying database connector uses for all the non-PDO based drivers. This doesn't happen with the PDO driver. This is also why SQL errors always display the full query on error screens.

If JLog were more PSR-3 structured I'd say that the query should be part of the context array and not displayed on screen in a message (but logged to the error log appropriately), but we don't have that option unfortunately as you can only give it a string to process (what's to be gained by showing the full query in an error message to the general public anyway?).

Closing this. Having the SQL query as part of the Exception message has been removed in 3.7, if you need the query in your error handling catch JDatabaseExceptionExecuting and call its getQuery method.