I think that defining the heritage of a group of people in political terms is always a fascistic idea. Commonly, we don't really do that.

To use a classic "muh Western huritage" example, when someone talks about ancient Greek heritage, they mean cultural heritage - food, music, art, theater, sport, architecture, religion, philosophy - and cultural values like democracy, patriarchy, social hierarchy including class-based slave ownership, etc. A specific ruling group, governing document, or even a specific war doesn't really belong in that category - those are a part of the history of that group, but not core to their identity as a group.

"Southern" US cultural heritage is made up of music (jazz, rock, blues, country, just to name a few), regional cuisines like BBQ & Cajun, maybe some historical ways of life that had a lasting impact - like cowboys & ranching culture, landed gentry like plantation owners, chattel slavery & black sharecropping, white sharecropping, Mexican settlers, etc., protestant christianity (specifically Baptist christianity) and cultural values like racial hierarchy and apartheid and its lasting impacts, an old-world sense of chivalry, and rugged individualism - stuff like that.

What doesn't really make sense is to say that the Confederacy itself is a core component of the culture. Not just because it only lasted for 5 years and non-white Southerners weren't willing participants, but also because the ideas/beliefs that shaped the Confederacy were a part of the culture of the most powerful group at the time. In other words, it was part of the culture of the South that allowed and enabled the Confederacy to exist, not the Confederacy that created the culture.