"Mister" (Mr) is a formal title, however, it isn't always used in a formal setting, nor does its use mean that the situation is a formal one.
Businesses often address their customers with a formal title and surname, eg "Mr Jones", "Mrs Jones" etc. However, this is changing in some kinds of businesses. For example, my bank asked me how I would like to be addressed when I first joined them, and they now call me by my first name. Many new businesses, especially those that attract younger customers, like to present a friendly image rather than a formal one.
Younger people are usually taught to address elders, particularly teachers, but also the parents of their friends with their title and surname. This is a mark of respect or recognition of authority more than a 'formality'. Some adults may, as a personal preference, insist that a younger person use their forename as an address.
More than ever, there seem to be fewer 'rules' about when to use, or not use this kind of address, and it seems to be dictated more by the personal preference of the individual. In your example, it sounds like Elena has been given just one name of the person she is meeting - "Bosley" - so she has assumed that (i) this is is surname, and (ii) that he must wish to be called by his formal name because that is the only name he has given. It turns out that it is his surname (you said his full name was 'John Bosley') but it seems he wishes just to be called by his surname as if it was his given name. This is not extremely common, but not unusual. Some people get called by their surname as a 'nickname', and it sticks. Other people choose to use their surname alone as a given name.
I would say the 'attitude' being shown by the character by using the name is a respectful one. She is approaching him for help, so is showing humility. She may view this as a 'formal' situation. As mentioned in the previous paragraph, it seems like she may also be showing respect by calling him what she believes he wants to be called. By saying "just Bosley" he is putting her at ease and making the situation somewhat more relaxed.