A project subname is a name a project gives out under its own name, like yourname.base.eth. The project runs it: you usually get it, renew it and change its records on the project's own site.
Where do I manage it?
On the project's own site. Registration, renewal and records are all handled there. If the project created your subname onchain on Ethereum Mainnet, in the ENS registry, you manage it in ENS Explorer instead.
Can I use it as my primary name?
A name ending in base.eth is already your primary name on Base from the moment you register it. Base sets that, it applies on Base only, and you can change it.
Setting an onchain Ethereum project subname as your Ethereum primary name in ENS Explorer is a feature still to come.
Where do I get help?
From the project that gave you the name. ENS only resolves it.
Want your own .eth name?
See Register a .eth name.