1.) Which WordPress Plugin, Template or PHP script can achieve it for me?
HOW IT WORKS
It is to be used on a Real Estate website. I am going to create Categories, Country, Region, States, Property Types and other custom Taxonomies, and then I will list Properties in them.
After this, I create a Section on the site (anywhere on the site) and I should be able to choose from existing Categories and Taxonomies, and the plugin would automatically list out the number of Properties in that Taxonomy (as you can see on the screenshot (https://prnt.sc/t51pf0).
2.) Secondly, you can see that when Agents and Agencies are registered, they are also treated as Post Types, and the Properties under them are also counted. See the screenshot https://prnt.sc/t51sr8
This means that the plugin makes Property count from all related and connected Taxonomies, and displays the number of Properties in connection to the Taxonomy.
3.) Thirdly, as Properties keep being added under any Category or Taxonomy, the number or count should change dynamically in Real-Time.
Which WordPress Plugin, Template or PHP script can achieve it for me?
Looking briefly at the site in your screenshot - it may appear visually like what you are after but it does not look like it is dynamically populating the information (if it is then it is doing it via a custom built feature), which is a long way from what you describe above
That said what you are after (if just using a list of different categories with quantity of posts) shouldn’t be that hard to build - although you will need the site setup to understand different properties and agents as posts.
Assuming your website is set up to populate properties and agents as posts then plugins like that (and still requiring custom work to create suitable layouts and connect everything) should act as a bit of a hacked solution if you don’t want to build it properly