Zell's the only orphanage member that was known to be adopted ( others could've been but parents were never met like Ma Dincht ) and he's likely the only one that made a choice to enter Garden and become a SeeD.
SeeD is his life ambition and entering some form of military came from being highly inspired by his late grandfather whom he greatly admires. SeeD is everything to him. An insult to SeeD is just the same as insulting him personally and will be met with the same response.
Despite all the backlash Balamb Garden in specific gets for its use of GF's and having cadets and SeeDs alike young teenagers seeing combat - Zell will adamantly and violently defend it.
Zell also canonly has the Garden logo tattooed on his shoulder at least by 17 years old. It's mostly covered up for obvious reasons, but it's there.