Granting permissions to a DB user - SQL Server
Reason you are not seeing the permission when you open the window by default it is scoped to Securable scope: Schema' and notdatabase`.

More on secureables.


  Can select permission on the database be visible from somewhere apart
  by executing T-SQL code?


Yes it is possible. From your 2nd screen:


Click Search...
Select radio button All objects of the types....
Click Ok
Select Databases


You should see TestDB in the list at top half of the window. In the bottom half you can see Explicit and Effective permission.

In case you want to use tsql:

EXECUTE AS USER = 'testuser';  
USE [DbaDatabase];  
SELECT * FROM fn_my_permissions (NULL, 'DATABASE');  
REVERT;
GO