What can a scanner do with my date of birth?

You asked - What can a scanner do with my date of birth - It depends on what other information they have. If it was in fact a scammer, they may have had other information but needed your birthday to make use of it. In that case, you'd have an imminent risk of identity theft if they were able to capture the additional information needed to make use of what they already had.

That said, name and birthdate by itself isn't really a big risk. I have my name, birthday, education, general area I live, e-mail address and home phone number all publicly listed on my website in easily scrapable fashion and have never had issues with identity theft. Most of that information can be obtained online already anyway from a variety of open source sources, such as social media and public records as well as from database dumps of compromised corporate databases which most people are likely to have been in. Certainly any expected fraudulent call is worth following up with the legitimate company to validate if it was real. If it was a real call, you have nothing to worry about. If it was not, then they will want to know so they can warn other customers if this kind of fraud is being targeted at many of their members.