BUG #18824: Inconsistent results for isolation level settings

The following bug has been logged on the website:

Bug reference: 18824
Logged by: dlyixue
Email address: 2322370369(at)qq(dot)com
PostgreSQL version: 17.3
Operating system: Ubuntu 22.04
Description:

When I was trying to set the isolation level, I realized that different
methods of setting the same isolation level gave inconsistent results.
This makes me wonder if the current setting is taking effect.
In my opinion, use case 1 and use case 2 should have the same results. Why
are the results of the queries different?
Is this a design problem? I feel like the documentation needs to explain the
difference as well.

The test cases are as follows:
Case 1:
reproduce=# BEGIN;
BEGIN
reproduce=*# SET SESSION CHARACTERISTICS AS TRANSACTION ISOLATION LEVEL
SERIALIZABLE;
SET
reproduce=*# SHOW TRANSACTION ISOLATION LEVEL;
transaction_isolation
-----------------------
read committed
(1 row)

Case 2:
reproduce=# BEGIN;
BEGIN
reproduce=*# SET TRANSACTION ISOLATION LEVEL SERIALIZABLE;
SET
reproduce=*# SHOW TRANSACTION ISOLATION LEVEL;
transaction_isolation
-----------------------
serializable
(1 row)