However, when I access the data through the Cache Management Portal, Cache Terminal, VSC, or IntelliJ, the Korean text appears broken, as I mentioned earlier.

Give an example of what you see in globals in the InterSystems Management Portal:
System Explorer > Globals

Since you have an 8-bit version and not Unicode, you probably won't see the hieroglyphs (최봉남), but rather something similar to (ÃÖºÀ³²).

This is not corrupted data, just a string stored in KSC5601/EUC-KR encoding.

From Caché terminal (Unicode version, locale = korw (Korean, Korea, Unicode)):

USER>w $zcvt("최봉남","O","KSC5601")
ÃÖºÀ³²
USER>w $zcvt("ÃÖºÀ³²","I","KSC5601")
최봉남

The same on Java:

System.out.println(new String("최봉남".getBytes(Charset.forName("EUC-KR")), Charset.forName("ISO-8859-1"))); // ÃÖºÀ³²
System.out.println(new String("ÃÖºÀ³²".getBytes(StandardCharsets.ISO_8859_1), Charset.forName("EUC-KR")));   // 최봉남

When I run below code in the Cache terminal, it returns 0, so it seems to be the 8-bit version.

That's what I thought. I have the Unicode version installed, the rusw locale, and there are no problems at all.

Class dc.a Extends %Persistent
{
Index is On s;
Property s As %String;

/// d ##class(dc.a).Test()
ClassMethod Test()
{
  s t=..%New()
  s t.s="최봉남"
  
  d t.%Save()
}
}

select * from dc.a

(When I check from the Management Portal, it shows as Latin1. If there's a more accurate way to verify this, please let me know.)

In SMP: System Administration > Configuration > National Language Settings > Locale Definitions
You should have something like kor8, enu8, eng8, etc.

Which version of Caché are you using: 8-bit or Unicode?
What locale is set by default?