Why is it "either . . . or" and "neither . . . nor"?

Do we say it this way because of some connection with French and the "ne . . . pas," "ne . . . ni" constructions? I'm thinking that it might be a direct importation from Old French by the Normans, or perhaps this duality is a general feature of PIE languages? If anyone can clue me in I'd be grateful.

I'm not quite sure what you're asking. Is it why we have negative-polarity terms for both arms of the conjunction? If so, that is something that many languages have, and certainly strikes me as natural, since both are negated.

Further, many (most?) languages allow you to insert as many negative words into a sentence as seems desirable: it's just English that has in the last few centuries acquired a bizarre rule that limits them. (Though in fact "I haven't got any" still has the negative-polarity "any", it's just not overtly negative).

'Neither...nor' and 'either...or' are correlative conjunctions. These work in pairs to join words and groups of words of equal weight in a sentence.
Nor – presents a non-contrasting negative idea
Or – presents an alternative item or idea

Logically therefore 'nor' means 'not or'.

So, if you don't have any bread or water to offer, you say:
'We have neither bread nor water.'
If instead you say,
'We have neither bread or water.'
it means you don't have any bread but you do have water.

Note: 'Neither...nor' is not a double negative.

From Old English:

na(w)ðer = nahwæðer = ne + hwæðer
neither = not + whether

e.g. We nabbað naðor ne hlaf ne wæter.
We have neither bread nor water.

Similarly, either is from Old English ǣgther, contracted form of ǣg(e) + hwæther = aye + whether.

Nor is a contraction of Old English nother, from neither, giving us:
We have neither bread, neither water.

'Or' is Middle English, a reduced form of the obsolete conjunction other (which superseded Old English oththe), and archaically was also used to mean 'either', giving us:
We have either bread either water.

So we say,
We have neither bread nor water
meaning
We have not either bread (and) not either water

If you say, 'We have neither bread or water', it means 'We haven't any bread or (but) we have water'.