Having a globally ordered, immutable, common ledger is a really useful thing in banking. Particularly in the inter-bank space where a vast quantity of data is shoveled around and everyone keeps partial, flawed copies.
Any given piece of data is, usually, required to be private to a small number of parties but is, usually, all required to be available to some parties i.e. regulators.
Doing all that is absurdly costly and error-prone. And that's when you're doing it well. Do it badly and the costs are eye-watering (ask, well, anyone).
Some of the ideas and techniques, particularly the cryptographic ones, used in the blockchains have opened the banking folks eyes to what could be possible. And this is a very active area of interest.
But it's not the bitcoin blockchain, per se, that's particularly interesting. For example, the trustless piece is largely a pointless waste in the problem I outlined above. We know all the participants and can safely move keys etc around. In the same vein, proof of work is unnecessary, we can identify block creators and easily use external remedies if they play silly buggers (no-one is suggesting we get rid of all our lawyers).
Now, you could argue that, if using the relevant techniques, you could keep all of that in a big, central DB and you'd be right. But it's not "big, central"-ness that's key. It's the "globally ordered, immutable, common"-ness. Along with selective privacy. Postgres doesn't give us what we want any more than Bitcoin does. The real change has been that we know relational DBs really well but we're learning about the possibility of the crypto and block chain tools. It's as much a mental shift as a technical one.
One final, minor point. A central DB, though possible, would be quite painful in practice. Although not fundamental, having a distributed ledger is actually quite handy. Although it would mean that we wouldn't completely do away with reconciliation hell, it wouldn't surprise me if something like a gold copy plus validated replicas approach or similar became popular. And again, some of the crypto techniques make this sort of thing much more tenable.