I find, in real life, I end up giving my full name or surname out frequently to taxi drivers, in fast food places, reservations, at interviews, when exchanging phone numbers/email/social media, etc. Some of my characters go by their surname instead of their forename (or both interchangeably depending on company) just as some people do in real life. I don't have full names planned for all my characters. But when it is more appropriate for the character to be addressed by their full name or a name that differs from the one I've been using in the narrative then I'll make one up then.
I occasionally introduce myself as Howard Bell in real life just because it's more memorable and impactful than my real name. Plus, if they get the reference we're generally automatic BFFs.
I would say it's strange, but it's up to you. I always give my characters full names. There's only two characters out of the probably hundreds I've had that I purposely didn't give a last name. With the first, nobody knows his real name, but he probably does have a full name. The second is just a figment of the protagonist's imagination.
