Should desktop apps display an information to the user after e.g. adding new customer to the database?
I see that some apps do this, some others don't.
Is this kind of information necessary to the user?
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UX Heuristic #1: Visibility of System Status applies here.
Whenever users interact with a system, they need to know whether the interaction was successful. Did the system actually catch that button press or was it busy with something else and it ignored it? Did the item get added to cart? Did the request go through? (One reason users have these questions is that they have been burned before by technology that didn’t work properly. However, even when the happy day of bug-free technology arrives, people will still wonder if they really clicked or tapped correctly.)
Appropriate feedback for a user action is perhaps the most basic guideline of user-interface design. It serves to keep users informed of the current status and to allow them to steer the interaction in the right direction, without wasting effort.
So yes -- generally, it is a best practice to let a user know that the task they were trying to complete, such as adding a user, was successful.
There are many patterns on how to do this, ranging from very subtle to necessarily intrusive. Here's another article on selecting the right one.
Adding to Izquierdo's answer, providing a summary for user-added information is one of the best practices for designing for users with anxiety:
We can reassure users by giving them the opportunity to check and change their answers before they submit. Without this step, users are less informed which could increase anxiety.
So yes, it is generally just a good practice.