I've never become acquaintances with a bank manager, but I have a lot of experience making acquaintances with the staff (managers included) at several restaurants that I frequent, and I believe that the same types of interactions can apply to your situation. The best relationship that I had with a restaurant staff was at a place we will call Restaurant A, where the manager would come over and personally welcome me back when I went there.
A few general thoughts
The thing that I think is the most key for building a relationship with your bank manager is frequency of interaction. When I was on good terms with restaurant A, it was because I went there nearly every day for a 4 month period while I was traveling for work. Employees in customer facing businesses such as restaurants and banks will see hundreds to thousands of customers a day depending on how busy they are. If you only see them sporadically, they are less likely to remember you. To start building a good impression with them, you need to have frequent interactions. It doesn't necessarily have to be a conversation with them either. For example, if you went into the bank several times in a short span, that might be something that a manager would take notice of, and may lead to them introducing themself to you.
Some specific things to do
Compliment the staff
One thing that managers in any sort of customer service work love to hear that customers appreciate their staff. The one thing that was the most helpful thing for me in building a relationship with the manager at restaurant A was that one night I asked my waiter if I could speak to him (the manager). When he came over I told him how excellent of service I had gotten at the restaurant and thanked him. Managers won't always remember the people who complain about their staff, but they will remember people who went out of their way to appreciate the work that their staff does.
Little interactions can make a big impression
Small talk is a great way to build a casual relationship with someone, especially in an employee/customer dynamic. Once you have had an initial conversation with the manager, make small talk when you see them. Something along the lines of
Good morning Susan, lovely day isn't it?
Hello Bob, how are the kids?
Notice in the second example, I included a question about the manager's children. Any sort of information that you can gather about the person is a good thing to slip into your small talk. Good topics include sports, their family, non-political current events, etc... Including things that the other person enjoys (even if you don't) in short conversations will show that you are taking a genuine interest in them. This is spelled out in a little bit more detail in the book How to Win Friends and Influence People (which is a good read, and I found it to be quite helpful). A good summary from the Wikipedia page:
Talk in terms of the other person's interest. The royal road to a person's heart is to talk about the things he or she treasures most. If we talk to people about what they are interested in, they will feel valued and value us in return.
Some closing thoughts
My previous point is predicated on already having met the person. That part, in my opinion, is the hardest. I personally like meeting the manager through complimenting their staff, because I find that it's an easy way to have a positive interaction with them.
