Blockchain (Bitcoin) as a database?
I'm very familiar with cryptocurrency and databases, and I can tell you it's not a great DB engine at all.  

Using the blockchain as a live database:

Think of it as a first normalized form without any really good built in search capability or indexing as far as the blockchain goes.  Basically a excel sheet without any computation capabilities that just gives you 'read/write' capabilities with lots of verification and validation.  A blockchain is a great way to validate your data is sanitized and correct before you put it in a database which let's you query it differently, index it, etc.  

Benefits of the blockchain:

The blockchain in this case is purely a ledger and an API for PUT and GET requests.  That's about it.  The blockchain is interesting because you need a majority of nodes to pass the transaction as valid and there aren't any rollbacks, once it's committed it's committed.  Thus if someone tries to put in a fake transaction it will be caught unless the person doing it has a pool which has a strong majority share.  Then they can validate it in their pool before someone can reject it.  That is the strong point of the blockchain.  Verification that the data is accurate.  It is also typically pretty slow.  You're looking at about 10 minutes under normal load for it to get validated.  Under heavy load the time goes up quite a bit.

After you have validated that the transactions are valid and not fraudulent using the blockchain, you can then import that data into a database and work with it however you like.  I have some experience with this but note that every single transaction on the current bitcoin architecture will be recorded thus it has some interesting info to analyze.  

Querying data out of the blockchain schema in a DBMS:

Here is the bitcoin diagram you can use to create the schema in PostgreSQL.  Using this you can then put it in a relational DBMS:
https://bitcointalk.org/index.php?topic=38246


This code repo is also helpful if you want to import the data into a real RDBMS:
https://github.com/bitcoin-abe/bitcoin-abe

As far as what DBMS you should put it in, that's up to your use case.  If you want to analyze the transactions/wallet IDs to see some patterns or do B.I. work I would recommend a relational DB.  If you want to setup a live ingest with multiple cryptocoins I would recommend something that doesn't need the transaction log so a MongoDB solution would be good.  I don't think you need to worry about Elastic Search unless you want to start doing live recording of all cryptocoins at the same time and will use it to do auto trading or something equally crazy. :)