The most common thing that happens in the case where you suddenly get a prompt to install WordPress when a site was running OK is that the database connection details were changed.
This can be caused by the host changing the database URL, or login or password was changed in the Cpanel; or the connection details were changed in the wp-config.php file, or even the wp-config.php file was deleted.
Check for the existence of the wp-config.php file using FTP, and if it is still there, check the connection details – database name, user, password, URL – with the Cpanel. See https://codex.wordpress.org/Editing_wp-config.php
The only strange thing is that there is the install.php installation
script into the admin folder and I am pretty sure to have removed
after the installation years ago.
This file will be reinstalled on an update; so the site was probably updated at some point in the past. (You can remove the file for security, but it’s not a great security defense.) An standard WordPress manual or automatic update won’t trigger the “install” screen; only an issue with wp-config.php and database details will do that.