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@qq.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)