It sounds like your phone is displaying your name on the app. Does it look like the email app itself "knows" your name or is it your phone? Send yourself an email. If your full name doesn't appear anywhere on the email, I would assume it is your phone that is displaying your name and not the app and I wouldn't worry about it. To be safe, you should also go through all of your email settings and options to make sure it still only "knows" your initials. I am like you and very private with things I do online and that is how I would test it.
Let us know what you find out so we can help others in the future that may have the same question.