Most banks likely subscribe to the misplaced belief that obscurity improves security.
As far as MySQL handling banking applications, I doubt it can scale that highly. I'm a pgsql guy, and I'm pretty sure it would have problems scaling that high as well. We're not talking dozens or hundreds of users at a time, but thousands upon thousands. Scaling up to those levels of usage while ensuring that you don't drop a 100th of a penny anywhere is another kind of application altogeher (chorus: it's another kind of application...)
MySQL's default behaviours would be very bad for an accounting database. It's tendency to just lop off parts of a value that don't fit, no check constraints, accepting column level FK syntax while ignoring it, allowing FKs from / to non-transactional tables, and tendency of replication to fail for various reasons with no obvious outward signs can all be worked around, but honestly there are far better choices for databases out there. Ones that don't require startup switches and carefully crafted DDL to make sure your foreign keys, check constraints, and input values actually work the way you'd expect it to.
MySQL is fine for holding non-critical numerical data, but honestly, if you're doing accounting on it, you're doing yourself a disservice using a tool poorly suited to the job.