Magento 2 Cache Issue – Page Content Not Refreshing

I have updated my response. In my previous answer, I had a block cacheable=false. This is incorrect because it do the page is not cacheable.

For knowing if the customer is logged or not, you must do the next:

Namespace/Modulename/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="Magento\Customer\Model\Session">
         <plugin name="namespace_module_customer_plugin_session" type="Namespace\Modulename\Plugin\Customer\Model\Session"/>
    </type>
</config>

Namespace/Modulename/Plugin/Customer/Model/Session.php

<?php    
namespace Namespace\Modulename\Plugin\Customer\Model;

use Magento\Customer\Model\Context as CustomerContext;
use Magento\Customer\Model\Session as CustomerSession;
use Magento\Framework\App\Http\Context;

/**
 * Class Session
 */
class Session
{

    /**
     * @var Context
     */
    private $httpContext;

    /**
     * Session constructor.
     *
     * @param Context $context
     */
    public function __construct(
        Context $context
    ) {
        $this->httpContext = $context;
    }

    /**
     * Check http context to know if user is really logged in
     *
     * @param CustomerSession $subject
     * @param bool $isLoggedIn
     * @return bool
     */
    public function afterIsLoggedIn(CustomerSession $subject, $isLoggedIn)
    {
        return $isLoggedIn ?: $this->httpContext->getValue(CustomerContext::CONTEXT_AUTH);
    }
}

When you do $customerSession->isLoggedIn(), it will return the correct value.

To retrieve the id of the customer, the best way is how says @SohelRana in its comment, using ajax or private cache. https://devdocs.magento.com/guides/v2.2/extension-dev-guide/cache/page-caching/private-content.html

Picking up from where you got stuck....
the implementation of the interface StateInterface is Magento\Framework\App\Cache\State.

method setEnabled looks like this:

public function setEnabled($cacheType, $isEnabled)
{
    $this->load();
    $this->statuses[$cacheType] = (int)$isEnabled;
}

This is in a default install. If you have other custom modules that use a different implementation for the Cache states interface maybe you should look in there.

Moving on to the persist method...
It should be in the same class

public function persist()
{
    $this->load();
    $this->writer->saveConfig([ConfigFilePool::APP_ENV => [self::CACHE_KEY => $this->statuses]]);
}

this is the actual method that should write in env.php.

You can easily test if when running the command you reach this method.
(use a simple var_dump, die for test purposes).

If you do not reach this code it means you are using a different implementation for this interface.
And you can check this easily by adding

echo get_class($this->cacheState);exit;

(or use xdebug and adding a breakpoint)
right before $this->cacheState->setEnabled($type, $isEnabled);