A “peer” is an equal, related to Latin “par.”
But a “peer” is also someone sho holds s title of nobility.
Does anyone know the story behind the development of these contradictory meanings?
Peers are at an equal societal level with other Peers.
Because the only peers that mattered were peers. Or at least it was at one point.
Because the people who were setting the terminology didn’t care about commoners. To them, you were either “important and special like me”, and hence a peer, or irrelevant.
Wasn’t the original term for nobles “peer of the realm”? So the original phrase specified what you were a peer of (the nobility/king) but common usage shortened the phrase and left the specification implied.
And not all nobles are Peers. Only Dukes, Duchesses, Princes, Princesses, and the Monarch are Peers, I believe, at least in the English system of nobility.
Being a Peer means you can talk to the Monarch as if you are their equal, in theory. In practice, not so much, but more so than anyone else in the realm.
There have been periods in English history where the Monarch was arguably less powerful than other Peers of the Realm. That was practically the default state in French feudal history.
I know a couple of other words that are their own antonyms, at least to a certain degree:
“cleave” can mean to cut apart or to adhere together.
“sanction” can mean to approve or to disapprove.
And a person they pee on is a peeee. 
Oh, and imflammable means flammable? Vat a country!
Actually, I’m wrong. A person that peers pee on is a peon.
I looked it up.
dracoi is correct. It’s Peer of the Realm, which has become contracted to Peer. And note, that as a title, it’s capitalised. In the House of Lords, a Peer is the peer only of their peers, the other Peers.
And to ‘bone’ (as in to ‘bone a chicken’) and to ‘de-bone’ both mean the same thing.
A couple of years ago, I read a book about the English language, and the author said something along the lines of (boy am I paraphrasing loosely here): “The English language tried to adhere to the rules of Latin, which is something like trying to play rugby by the rules of baseball. . .” 
And likewise, in reverse, for dusting something: are you removing a light powder or coating stuff with one?
Ravel, and by extension unravel, are both their own antonyms.
As are ‘peel’ and ‘un-peel’.
Gosh, I could play this game forever! Are you sure we’re in the right forum?
Words that are their own antonyms are sometimes called contronyms or auto-antonyms: See the Wikipedia page and their list of contronyms in English. “Peer” is not currently on their list.
“Thaw” and “unthaw” mean the same thing.
Ditto “ravel” and “unravel.”
That’s a stylistic issue. And your reasoning is wrong. You’re not using it as a title. That would be – “Peer John Smith knelt before the queen.” And as far as I can tell it’s not commonly used that way. In “John Smith is a peer of the realm,” there’s no need to capitalize.
How about “sanction”, almost. To sanction something means to condone or permit it. But “sanctions” against someone or something are punitive restrictions.
“Fast” – rapid
“fast” – unmoving
“Cleave” can mean either join together or split apart.
“Moot” can mean “subject to debate” (“moot court”) or “undebatable” (“moot point”).
Language is not logical.