I think you have not added the newly installed PostgreSQL16 to the pgAdmin 4. I have attached the docs link here for your reference to add the server to pgAdmin 4.
I have installed PostgreSQL 16.3(the latest version) and followed the steps as required. Once the setup is done, the popup is closed.
After that, I opened the pgadmin4 to connect with the PostgreSQL but was unable to find it under the dropdown of servers and it does not ask for the password as well which i have created earlier to connect with that.
And I have provided the binary path as well but it does not work.
Please find the attached screenshot of it and kindly help me to resolve it so that i can practice without any issue.