I regard “scholar” as one of those descriptors that are conferred by others and not assumed for oneself. It’s more than just a label or an earned degree; it expresses honor and respect.
Others that I see as similar include hero, saint, philosopher, sage, artist, holy man or woman, and wise man or woman. A person might say “I’m a soldier,” for instance, but I don’t expect him to say of himself “I’m a hero.” If she tells me “I’m a painter,” that’s fine, because she’s just telling me what she does; but if she says “I’m an artist,” I think she’s making an immodest evaluation of the quality of her own work.