This question's purpose has been narrowed down and is more specific. The author, from what I can tell, is requiring assistance with Magento 2's review/summary section of the checkout and acquiring the customer data needed to output that information. If I had to guess, the rift is between the data and how it is provided (requireJS, checkout and customer data JS stuff, knockout templates) and the templates to render it. Totally out of my current knowledge area.
What I have written below should not be followed, as I had assumed that I was more familiar with an area of Magento 2 than I actually was. This is something I will be more careful of in the future. The purpose of me leaving this is because the original question has been clarified quite well, in my opinion. The author should be able to find an answer from people that have the experience and knowledge to do so.
From what I understand, communicating with the author has narrowed down the question to:
How do I update the theme and settings in M2 checkout summary/review page to match the screenshot. An emphasis seems to be on producing the template code behind display of
- credit card details,
- address (billing, assumed by screenshot)
- cart items
Review the configuration settings for checkout:
Step 1: Configuration checks. In the checkout options, be sure to configure them to suit the requirements.
Step 2: Update theme layouts / templates, as needed.
Review the Frontend Developer Guide: (http://devdocs.magento.com/guides/v2.2/frontend-dev-guide/bk-frontend-dev-guide.html) and particularly Templates customization walkthrough (http://devdocs.magento.com/guides/v2.2/frontend-dev-guide/templates/template-walkthrough.html)
It is important to understand how layouts work in the event they were customized (http://devdocs.magento.com/guides/v2.2/frontend-dev-guide/layouts/layout-overview.html)
The review page template from the Luma or Blank theme should show the proper code, so I am hoping that an update of any necessary templates in your custom theme wouldn't be an issue if you use code referenced from those default review templates.
If you do not have templates in your theme, please copy the proper template file to your theme. If you have issues displaying the review template, check that the layouts xml files are not modified to remove or update the template file used.
If there are any specific items missing from there, search for a question answering it, or specify which exact ones in a new question targeting the item. There is a chance that this may be a question to help customize the checkout review page with some additional information not available in the default theme hierarchy.