Tagging PostgreSQL database
As a_horse commented, a COMMENT ON DATABASE db IS 'foo'; would be a simple option. 

set_config() settings are for the duration of the transaction or session only. If you need to persist information beyond the current session, that won't do.

For more structured bookkeeping, maybe including a timestamp etc, consider a bookkeeping table in another database in the same db cluster - without altering the db in question at all. (Maybe in addition to a comment on the main db indicating the current status.) Sample psql session:

CREATE DATABASE db_meta;
\c db_meta
CREATE TABLE db_status_log (
  log_id serial PRIMARY KEY
, log_ts timestamptz NOT NULL DEFAULT now()
, status text
);


Etc.

\c db_meta is the psql meta-command to connect to a db.