Yes to both of your questions. Both are those are feasible. At least, the second one is certainly feasible; to the extent that I understand the first scenario, I believe that's possible too.
There are even existing systems that do something like this. For instance, Namecoin is a system for registering domain names that uses a Bitcoin-like blockchain. Only one person can own a domain name at a time, so it has the "uniqueness" aspect you are getting at.
One could easily support a system where a central authority mints digital tokens, e.g., to indicate who has a particular trading card. Then, digital tokens can be traded online, using a Bitcoin-like blockchain to record the transactions.
The general term of art for this broad area is altcoin or cryptocurrency. There are many Bitcoin-inspired cryptocurrencies: see Wikipedia's list or this list.
From a fundamental computer science perspective, Bitcoin solves the distributed consensus problem: it records a sequence of transactions, ensures that these are consistent according to some rules (e.g., no double-spending, can only spend coins that are currently yours), enables anyone to add additional transactions, and ensures that everyone in the world sees the same view of the database of transactions.
However, there are some non-trivial challenges to extending Bitcoin to your particular case. Bitcoin is a sophisticated mechanism that relies upon a clever allocation of incentives. The security of Bitcoin relies upon people to do "mining", as this is what enables appending to the blockchain and prevents forking of the bitcoin. Bitcoin creates an incentive for mining through its proof-of-work-based "lottery": miners who successfully add a new block to the end of the blockchain are given a small reward, a small fraction of a Bitcoin. This works because Bitcoin is used for a currency. If you used Bitcoin for, e.g., centrally-assigned trading cards, it's not clear how you would design the system to provide an incentive for mining. So that aspect of the system design would need careful thought and consideration.