Postgres: "secondary" primary key

How would one have dynamic "secondary" keys? For example, how we have it in Jira. We have a project name, for example "Test" with the shortcut "TEST". All associated tickets in this project are given a ID "TEST-1234", "TEST-1235", and so on.

How would one model this in Postgres?

In my mind, I worked out a table project with name VARCHAR(xx) and short VARCHAR(8). The ticket table would somehow look like this:

CREATE TABLE ticket 
(
    id INTEGER PRIMARY KEY AUTOINCREMENT,
    project_id INTEGER NOT NULL,
    identifier INTEGER ???,
    FOREIGN KEY (project_id) REFERENCES project(id),
)

How would you construct an identifier that is incrementing for every new ticket that is referencing the same project_id?

These are examples from the older thread I suggested, adjusted to your schema:
_{demo1 at db<>fiddle}

create table ticket_id_per_project(
  project_id INT PRIMARY KEY REFERENCES project(id),
  max_ticket_id INT NOT NULL DEFAULT 1);

create function f_ticket_ids()returns trigger language plpgsql as $f$
begin
    insert into ticket_id_per_project as f
    values (new.project_id,1)
    on conflict (project_id) do update
    set max_ticket_id=f.max_ticket_id+1
    returning format('PROJECT_%s_TICKET_%s',project_id,max_ticket_id)
    into new.identifier;
    
    return new;
end $f$;

create trigger t_ticket_ids before insert on ticket
for each row execute function f_ticket_ids();

It makes no attempt at removing gaps or correcting manually broken sequences - if you remove a ticket, you're left with a gap there. Also, all concurrent insert into ticket attempts for the same project, have to wait for predecessors to finish - the delay is negligible but non-zero. Example with brand-new projects and fresh tickets:

with gen_projects as (
  insert into project 
  select from generate_series(1,3) 
  returning *)
insert into ticket(project_id)
select id from gen_projects
returning *;

The ticket counter continues for each existing project individually:

with pick_projects as (
  select id from project tablesample bernoulli(50)repeatable(.42)limit 2)
insert into ticket(project_id)
select id from pick_projects cross join lateral generate_series(1,3)
returning *;

You can trade that for only locking if they all insert the first ticket for a given project. Otherwise, concurrently inserted tickets for the same project don't block each other:
_{demo2 at db<>fiddle}

create function seeded_sequence_nextval(seed text) returns int as $f$
declare sequence_name text:='seeded_sequence_'||seed;
begin 
  begin execute format('create sequence if not exists %I;',sequence_name);
  exception when /*23505*/unique_violation then null;--concurrent create if not exists
  end;
  return nextval(format('%I',sequence_name));
end $f$ language plpgsql;

create or replace function f_ticket_ids()returns trigger language plpgsql as $f$
begin
    select format( 'PROJECT_%s_TICKET_%s'
                  ,new.project_id
                  ,seeded_sequence_nextval('tickets_project_'||new.project_id)) 
    into new.identifier;
    return new;
end $f$;

The function was swapped in-place, so there's no need to redefine the trigger. Works just the same, except for the locking.