A Facebook Page or Place may exist for a business even if the business didn't create it. This can occur for a variety of reasons. I think the most common reason is when a person checks into a location that doesn't already exist, they can create a new Page to represent the location. Many of these pages will be labeled as a "community page" meaning it doesn't officially represent the topic. I see this a lot with bank branches when employees or customers "check in" to a branch and end up create a new page for the location.
You have several options with these types of pages - 1.) you can request to claim the page, 2.) you can report the page and attempt to have it removed, 3.) you can merge the page with an existing page (after it is claimed).
Regarding personal accounts and Facebook pages - I believe it is a violation of Facebook's terms and conditions to create a false presence to manage an organization so I don't recommend created a fake account to mange the bank's page; however, if the bank's Facebook Page is in fact an official Page, then you can have multiple "Administrators" of that Page (managed under Settings > Page Roles). Various roles can then be assigned to one or more valid Facebook accounts to manage the bank's Facebook Page. Facebook Page Admins can log into their personal Facebook account and change what account (personal or Page) they are managing. This can be confusing and has caused people to post business items under their personal profile or personal posts as if it were the bank, so I recommend Administrators be trained. This is also a good reason to consider Social Media Management solutions.
I hope this helps some.
Thanks,
Russ