Is an academic degree an honor, a title, or both? For example: Do graduates receive a Bachelor of Arts degree or become a Bachelor of Arts?

Are Bachelor degrees grammatically and/or etymologically distinct from advanced degrees?

I note that the titles Master and Doctor are still in common usage, even though in practice frequently divorced from the associated academic degrees.  (E.g, a Master as a formal designation is now most frequently associated with a skilled trade – like master electrician.  And many holders of doctorate degrees eschew the use of the title outside of medicine and the academy.)

Etymologically it appears that Baccalaureate would be the correct term for a person who has received a Bachelor's degree.  But I can't find that in modern usage, and its etymology (laureate) emphasizes the award of the degree, rather than the achievement of mastery or doctoral skill.

Or do I have it backward, and it is modern academia that has appropriated these different titles without establishing such terminological consistency?
Depends on the country and education system. 

For example, in English, 'bachelor' doesn't mean anything but that you finished that level of education.  In Spanish, finishing university doesn't mean a person gets recognized as having the degree: there are further methods to achieve that (titulaciòn), after which the government officially recognizes the degree and the title that comes white it like 'ingeniero' (engineer) or 'abogado' (lawyer). That is why in some countries the degree is added as a title at the start of a person's name.