Hat trick
According to etymonline, ‘hat trick’ comes from:
1879, originally from cricket, "taking three wickets on three consecutive deliveries;" extended to other sports (especially ice hockey) c.1909. Allegedly because it entitled the bowler to receive a hat from his club commemorating the feat (or entitled him to pass the hat for a cash collection), but also influenced by the image of a conjurer pulling things from his hat (attested by 1876). The term was used earlier for a different sort of magic trick:
Place a glass of liquor on the table, put a hat over it, and say, "I will engage to drink every drop of that liquor, and yet I'll not touch the hat." You then get under the table; and after giving three knocks, you make a noise with your mouth, as if you were swallowing the liquor. Then, getting from under the table, say "Now, gentlemen, be pleased to look." Some one, eager to see if you have drunk the liquor, will raise the hat; when you instantly take the glass and swallow the contents, saying, "Gentlemen I have fulfilled my promise: you are all witnesses that I did not touch the hat." ["Wit and Wisdom," London, 1860]
Not really a very interesting etymology at all—not much enlightenment to be found, I fear, just a silly cricket tradition with a hat.
Brace
‘Brace’ is more interesting to me. First of all because I didn’t know scoring two goals in a match was called a brace, or anything at all for that matter; but also because it’s more of a ‘real’ etymology.
The word comes from ‘a brace’, which is an old term of venery. It means ‘a pair of [some animal, especially birds] caught in the hunt’. So if you go out looking for dinner, shoot two pheasants, and bring them home, you’ll have a brace of pheasants. It makes sense, then, that a word that means ‘a pair’—in particular, a pair of things that you have managed to obtain through skill—would be a good word to use for scoring two goals in a football match.
The word ‘brace’ itself is related to the verbs ‘to brace’ and ‘to embrace’: the original meaning (from Old French brace) is ‘[a pair of] arms’. ‘To brace’ originally meant ‘clasp/fasten tightly’, which is a plausible meaning for a verb directly derived from the word for an arm. ‘To embrace’ is simply to put something in [your] arms.
Old French brace is quite straightforwardly the regular outcome of Latin bracchia, the plural of the Latin neuter word bracchium ‘arm’, itself a loan from Greek βραχίων ‘arm’. The Greek word is interesting in itself, since it is simply a nominalised form of βραχίων, the comparative of βραχύς ‘short’ (found in English words as the prefix ‘brachy-’). Quite possibly, the word originally meant ‘upper arm’, being then called literally the ‘shorter arm’ (as opposed to the longer lower arm).
Βραχύς, in turn, is related to Latin brevis ‘short’, which (through Old French again) shows up in English as ‘brief’, as well as to some other, rather obscure words (there’s an old Gothic verb gamaurgjan that means ‘to shorten’).
Best Answer
The word identity is being used here in something like its mathematical sense. That is to say that thought and being are inextricable, that they are in essence the same thing, that human thoughts are in some sense governed by the condition of being human animals who interact with the world around us with human senses and human means of locomotion, et cetera.
Identity, in this context, means sameness. In mathematical (and computing) use, things share an identity if they are precisely the same thing. Adding or subtracting zero, or multiplying or dividing by one are identity operations.
In common usage, we are often referring to a metaphorical "on paper" or conceptual representation of ourselves when we use the word identity. It isn't exactly who we are, but who we think we are, who others think we are, or the distorted picture of us created by the trails we leave behind.