As long as the abstract DatabaseConnection class extends PDO class, you can invoking pdo methods in order to know the current database driver.
For instance:
$conn = Database::getConnection();
print $conn->getAttribute($conn::ATTR_DRIVER_NAME); #returns mysql, pgsql...
There is a second way to do it using DatabaseConnection::driver():
print $conn->driver();
or DatabaseConnection::databaseType();
print $conn->databaseType();
Note that DatabaseConnection::driver() and DatabaseConnection::databaseType() are similar functions but not equals!
The return value from DatabaseConnection::driver() method depends on the implementation and other factors.
in the Drupal Database API page:
database.inc abstract public DatabaseConnection::driver()
This is not necessarily the same as the type of the database itself. For instance, there could be two MySQL drivers, mysql and mysql_mock. This function would return different values for each, but both would return "mysql" for databaseType().
In the most cases you just gonna want to use only
$conn->getAttribute($conn::ATTR_DRIVER_NAME)
or $conn->databaseType()
If you want get more specific properties, you should take advantage the PHP ReflectionClass features:
$conn = Database::getConnection();
$ref = new ReflectionClass($conn);
#ref->getProperties, ref->getConstants $ref->isAbstract...
Reference:
PDO::getAttribute
PDO::ATTR_DRIVER_NAME
Drupal Database API
Drupal Base Database API class
select version();