I would suggest {server name}/{db name} as name instead of did is better for UX. You can get the db name from conn object and server name using the sid (refer - web/pgadmin/browser/server_groups/servers/__init__.py)
I have been working on #4943 ("Database connected" success message itself is not enough) added by you on pgAdmin4 issues.
I want to know what details of database should be included in success message.(i have added 'did')
The success message can be modified by altering json response in the connect method in file PGADMIN_SRC/web/pgadmin/browser/server_groups/servers/databases/__init__.py.