Can a trigger procedure in PostgreSQL learn its own name?
You can query pg_trigger catalog table and gather this information:

SELECT
    tg.tgfoid::regproc::text
INTO
    v_function_name
FROM
    pg_trigger tg
WHERE
    tg.tgrelid = TG_RELID
    AND tg.tgname = TG_NAME;


v_function_name is a variable you must declare before (as text or varchar).

Casting to regproc and then to text will make it double quote the name if not simple (just like quote_ident), if you want to avoid it, you can simple join pg_proc and get proname (in this case v_function_name can be of type name):

SELECT
    pr.proname
INTO
    v_function_name
FROM
    pg_trigger tg
    INNER JOIN pg_proc pr ON pr.oid = tg.tgfoid
WHERE
    tg.tgrelid = TG_RELID
    AND tg.tgname = TG_NAME;