It does sound too formal for social use. You could say 'What shall I call you?' or 'What would you like me to call you?', but in reality, you would be unlikely to have to ask this. As you know, English doesn't have a system of grammar connected to status and hierarchy in the way that Japanese does, or specific forms of address to indicate relationships between people. In fact, the only issue is whether to call someone by their first/given name, or whether to address them as Mr/Mrs.Ms and their surname. With a friend of a friend, you would automatically use their first name. With the parents of a friend, you would take the 'cue' from them. Your friend's mum might say 'Hi, I'm Judy', in which case that's what you'd call her. Otherwise it would be polite to call her Mrs Brown (or whatever), after which she might say 'Call me Judy', and then you'd use her first name.

Listen to how the person introduces themselves. Usually you can go by the name they say, as su.ki. said. If someone states their first and last name I would normally use their first name still outside of certain circumstances. A younger student will not call their teachers by first name normally, in University or College it can be more acceptable. If you are unsure when you are given the first and last name, just ask "Can I call you.."