The mafia analogy works really well if you play in a Camarilla-led city, but my group just got the Prince ousted to turn it into an Anarch domain and to be completely honest, i just play V:tM the same way i play Thirsty Sword Lesbians, except that i constantly make puppygirl jokes about the Gangrel player's character and really lean into the "Tremere are a trans metaphor" angle.
Edit:
Also some minor note on treating clans as internal groups within the Camarilla: This only really works if you play in a really large city with tons of vampires and handwave the masquerade breach potential of having hundreds of bloodsuckers running around town. Our campaign runs with the suggestions in the book and has a total sprawl-wide vampire population of 20-30 (we play in Hamburg, it's just nowhere near as huge and chaotic as LA or Mexico City and the lick population reflects that). You can't have a dozen internal divisions at that size. Two main factions (in our case, Camarilla and Anarchs) and a loose cannon here and there (like a former Sabbath shovelhead leading the local Bahari, which are mostly her ghouls and other humans) are much more likely at that scale, and anything else are outside forces like the 2nd Inquisition, Lupines or out of town vampires who are on the run from the purges in the UK and try to gain a foothold at the cost of established vampires. The only Tremere in the city are my blood witch and sometimes her sire when she isn't exporting revolutiion elsewhere, and a lot of clans just do not show up in our chronicle at all. Ventrue are larger and mostly the Venn diagram between them and the Camarilla is a circle, but the other clans do not have elaborate family structures and internal workings, it's usually just a sire, a childe and maybe one other person.
that gamers needed this spelled out for them