It's the same commander. You'd just choose Open, PG, Solo, or a private modded server when you log in.
It's not working like that.
You should watch Dav's show (from 2015?)when he presented the online infrastructure (the server side) behind Elite
Anyway, as per OP, a private server would imply it's own galaxy, it own accounting - it would be pure private and not related to FDev in anyway.
So you wont be logging in into Elite on Frontier, but into Elite on some random dude private server hosted in his flat not in AWS
And as a side track:
You said you played DAoC.
You knew that private devs somehow reverse-engineered the server-code of DAoC servers and then recoded it to obtain an open source DAoC Server and so the Freeshards appeared.
No more paying monthly subscriptions to Mythic (we in Euro zone, we payed subs to GOA)
Your DAoC client could connect to a freeshard an play for free, completely unrelated to Mythic/Goa - sure under their freeshard ruleset and with the reserve that your toon could vanish at any time - but it was free so who cared?
I tried a such Freeshsrd back then - it was stuck at SI level, but still fun, However they had a strict policy of no buffbots - so i gave them the finger and found something else to play (Conan IIRC

)
The moment FDev give away their server code to players to host their ED private servers, then opensource ED servers would appear and...