This post's title refers to ODO's definition 1. Used to emphasize something surprising or extreme.
5. Exactly, precisely, just. In later use generally somewhat archaic.
8. a. Used to convey that what is being referred to is an extreme case in comparison with a weaker or more general one which is stated or implied in the adjacent context. Prefixed to the particular word, phrase, or clause in which the extremeness of the case is expressed.
b. In uses parallel to those at A. 8a, but placed after the word, phrase, or clause to which it relates.
[ Etymonline : ] The adverb is Old English efne "exactly, just, likewise." Modern adverbial sense (introducing an extreme case of something more generally implied) seems to have arisen 16c. from use of the word to emphasize identity ("Who, me?" "Even you").
Etymologists are uncertain whether the original sense was "level" or "alike."
Q1. What semantic notions underlie 'levelness' or 'alikeness', with 'something surprising or extreme'?
Q2. Don't the semantic notions of 'levelness' or 'alikeness' contradict the notion of 'something surprising or extreme'? If X is more extreme than Y, then X and Y are neither level nor alike.