yeah, I can't relate to this at all either... even after decades of being exposed to the anime and manga. Maybe it's like some not-so-close acquaintance calling you by your nickname? But for me it's not embarrassing though, just awkward.. Hmmm
@emmJH@Tensei21 speculating here, but I imagine it's the cultural equivalent (well, similar at least, cause they do allow other close friends to use it) of a 'pet name' that partners have for each other in (some) western culture? So, the embarrassment is not just them using the name, but the closeness it implies, as well as the reciprocal feelings.