Magento 2 – Writing Messages to a Log File

You do not need to customize or try to extend Magento2's logging. As you said it's using Monolog with only slight customization. It is sufficient to write your own logger extending Monolog with very little effort.

Assuming your module is in YourNamespace/YourModule:

1) Write Logger class in Logger/Logger.php:

<?php
namespace YourNamespace\YourModule\Logger;

class Logger extends \Monolog\Logger
{
}

2) Write Handler class in Logger/Handler.php:

<?php
namespace YourNamespace\YourModule\Logger;

use Monolog\Logger;

class Handler extends \Magento\Framework\Logger\Handler\Base
{
    /**
     * Logging level
     * @var int
     */
    protected $loggerType = Logger::INFO;

    /**
     * File name
     * @var string
     */
    protected $fileName = '/var/log/myfilename.log';
}

Note: This is the only step that uses the Magento code. \Magento\Framework\Logger\Handler\Base extends Monolog's StreamHandler and e.g. prepends the $fileName attribute with the Magento base path.

3) Register Logger in Dependency Injection etc/di.xml:

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" 
    xsi:noNamespaceSchemaLocation="urn:magento:framework:ObjectManager/etc/config.xsd">
    <type name="YourNamespace\YourModule\Logger\Handler">
        <arguments>
            <argument name="filesystem" xsi:type="object">Magento\Framework\Filesystem\Driver\File</argument>
        </arguments>
    </type>
    <type name="YourNamespace\YourModule\Logger\Logger">
        <arguments>
            <argument name="name" xsi:type="string">myLoggerName</argument>
            <argument name="handlers"  xsi:type="array">
                <item name="system" xsi:type="object">YourNamespace\YourModule\Logger\Handler</item>
            </argument>
        </arguments>
    </type>
</config>

Note: This is not strictly required but allows the DI to pass specific arguments to the constructor. If you do not do this step, then you need to adjust the constructor to set the handler.

4) Use the logger in your Magento classes:

This is done by Dependency Injection. Below you will find a dummy class which only writes a log entry:

<?php
namespace YourNamespace\YourModule\Model;

class MyModel
{
    /**
     * Logging instance
     * @var \YourNamespace\YourModule\Logger\Logger
     */
    protected $_logger;

    /**
     * Constructor
     * @param \YourNamespace\YourModule\Logger\Logger $logger
     */
    public function __construct(
        \YourNamespace\YourModule\Logger\Logger $logger
    ) {
        $this->_logger = $logger;
    }

    public function doSomething()
    {
        $this->_logger->info('I did something');
    }
}

You should not install Magento on a top-level domain or IP address: https://github.com/magento/magento2/issues/2439