In simplest terms, Islamic principles need to apply to all aspects of life and government. Which means the principles of Islamic finance need to apply to all financial transactions, regardless of whether they're conducted by a retail bank, a commercial bank or a central bank.
In practice, the rules that apply to one institution vs another could differ in significant ways only because the nature of transactions themselves are fundamentally different. Most of the rules that laymen are aware of when they consider "Islamic finance" are explicitly designed around retail banking and personal transactions, since that's all they're expected to be dealing with. But even if all the rules are built upon the same principals you can't just copy-and-paste them across the board.
In general, the principles that are agreed upon are that no transactions can contain elements of:
- Riba (usury)
- Maysir (speculation, gambling)
- Gharar (uncertainty, excessive risk)
In addition, there may be other Islamic principles involved that rarely come up in small scale personal transactions but which could apply to a regulatory body such as the central bank; maslahah (public interest) being a particularly common one.
The modern financial system really is nothing like anything in place during the time of the prophet, so there is still a lot of discussion and dispute about exactly what these fundamental principals of Islamic finance even mean and how or when to apply them to modern practices. But the general idea is that yes, they always apply, even if they're not always relevant.