How to add extra container to page header in Magento 2

Add a container after="header.panel.wrapper" inside header.container. By the reference of the container add your template.

Inside your default.xml

<?xml version="1.0"?>
<page xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:View/Layout/etc/page_configuration.xsd">
    <referenceContainer name="header.container">
        <container name="header.containertwo" as="header_containertwo" label="Page Header Container"  htmlTag="header" htmlClass="header-mini-container" after="header.panel.wrapper"/>
    </referenceContainer>
<referenceContainer name="header.containertwo">
    <block class="Magento\Framework\View\Element\Template" name="header.mini.container" template="Vendor_Module::test.phtml"/>
</referenceContainer>
</page>

In the test.phtml

<div style="border:1px solid red;" id="test123"><p>This is a test</p></div>

Out put

Sample module

/app/code/Vendor/Module/registration.php

<?php

\Magento\Framework\Component\ComponentRegistrar::register(
    \Magento\Framework\Component\ComponentRegistrar::MODULE,
    'Vendor_Module',
    __DIR__
);

/app/code/Vendor/Module/etc/module.xml

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="../../../../../lib/internal/Magento/Framework/Module/etc/module.xsd">
    <module name="Vendor_Module" setup_version="1.0.0" schema_version="1.0.0">
    </module>
</config>

/app/code/Vendor/Module/view/frontend/layout/default.xml

<?xml version="1.0"?>
<page xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:View/Layout/etc/page_configuration.xsd">
    <referenceContainer name="header.container">
        <container name="header.containertwo" as="header_containertwo" label="Page Header Container"  htmlTag="header" htmlClass="header-mini-container" after="header.panel.wrapper"/>
    </referenceContainer>
<referenceContainer name="header.containertwo">
    <block class="Magento\Framework\View\Element\Template" name="header.mini.container" template="Vendor_Module::test.phtml"/>
</referenceContainer>
</page>

/app/code/Vendor/Module/view/frontend/templates/test.phtml

<div style="border:1px solid red;" id="test123"><p>This is a test </p></div>