tl/dr: Passwords are like old food: when in doubt, throw them out. If you're worried just change your password.
3rd Party Financial Services
First some background: the banking industry can, unfortunately, be a bit behind the times in this area. As a general rule of thumb there isn't yet a great way (at least in the US) for 3rd party financial services (such as Emma) to easily integrate with banks. I'm sure that some banks have well-secured API services for just these sorts of things, but many banks (especially smaller ones), just don't.
As a result it is unfortunately common for 3rd party financial services to access your account by asking for your actual online banking username and password. It's a terrible idea, and it really shouldn't be done like that, but unfortunately it is.
One (of the many) reasons why it is so dangerous is because it teaches people to hand over their usernames and passwords. When you have a service you integrate like this and then use it successfully for a while, it makes it seem less crazy when a scammer comes around and says, "I just need your username and password". Anytime someone asks for your username and password to your online banking the default response should be exactly what you did - "NO WAY! You're crazy!". Unfortunately though, the fact that a service asked for your username/password doesn't actually mean that it is a scam, because in many cases that is the only way that legitimate 3rd party financial services can actually access your data.
Unfortunately telling the difference between "Legitimate and safe 3rd party service", "Legitimate but unsafe 3rd party service", and "Outright scam" can be very difficult or just impossible. That leaves a lot of people in your boat.
Can you leave your account partially registered?
This is really up to you. If you signed up with a strong password then you probably have very little to worry about. Personally though I would reach out to their customer service and ask them to delete my account. You may or may not have success.
Password in Username
Only Emma would be able to tell you the implications of accidentally sending your password in the email field. Personally I wouldn't be too concerned. It is possible that they log all details of all failed login attempts, and therefore there is a plain-text copy of your banking account password somewhere in their system. From my own personal experience though I don't expect many companies do this, so the risk is likely low. Either way an employee seeing that in the logs, fetching your bank account info, and trying to login to your bank account to do something malicious is probably also very unlikely.
Still, think of this as a risk/benefit analysis. The risk of someone seeing your password and trying to log into your bank account is hard to predict but probably low. However, the cost of changing your online banking password is probably also very low, and the benefit of doing so is that you no longer have to worry about whether or not someone will use your old password. As a result, if you're worried, just change your online bank account password.