Learn English – Why “a” bow and arrow

The expression “bow and arrow” is an example of siamese twins, also known as irreversible binomials, binomials, binomial pairs, freezes and nonreversible pairs in linguistics. There are several posts on EL&U that talk about this language feature which is not exclusive to English.

Binomials have two main characteristics. The first, as noted by the Original Poster, is that the order is usually perceived as fixed. The second is that the two terms are normally the same part of speech, though not always Nouns.
^Araucauria

Frozen reduplicative phrases like these, especially ones made of nonsense or phonosemantic roots like riffraff or hocus-pocus, are simply called Freezes in the literature, following Cooper and Ross 1975, the first study to investigate them thoroughly.
^{John Lawler}

Thus ‘word-pairs’ such as bed and breakfast; birds and bees; cat and mouse; Adam and Eve; fish and chips, and trinomials such as: blood, sweat and tears, and lock, stock and barrel are clichés whose word order is normally fixed.

Using the OP's example, people rarely say or write: “arrow and bow”; “arrows and bows”; “bows and arrows”; or “a bow and an arrow”.

...the more general tendency to represent powerful groups first (Hegarty et al., 2010). This explanation is in line with Cooper and Ross's (1975) and Mollin's (2010) observations derived from language corpora that more powerful elements are mentioned before less powerful ones (as in bow and arrow; sun and moon; parent and child; cow and calf) and with McGuire and McGuire's (1992) argument that groups of higher sociocultural status are mentioned first.
^{The Oxford Handbook of Language and Social Psychology}

See also the recent EL&U question Why do we say ‘kith & kin’ and not ‘kin & kith’?

However, bow and arrow is quite unique, it ought to be bow and arrows, inasmuch as the first noun is singular (because we use ‘a bow’,) and its corresponding partner, arrow(s), should be plural because an archer uses more than one arrow to shoot. Instead, both terms are singular, they seem to follow the unspoken rule that pair nouns in binomials remain either singular or plural, such as: mother and child; chalk and cheese; friend and foe; hammer and nail. Or plural as in: brothers and sisters; birds and bees; ladies and gentlemen; and odds and ends.

Occasionally, the first noun is singular or uncountable whilst the second is plural;
e.g. fish and chips; bacon and eggs; and fun and games.

But even rarer are the instances where the first noun is plural followed by a singular or uncountable noun: strawberries and cream; pickles and ice cream, were the only examples I could come up with.

According to Cooper and Ross (1975:65-66) cited by Susan Mollin (The (Ir)reversibility of English Binomials...)

There are twenty semantic principles [the authors] distil from their collection of freezes: […] adults before children, male before female, positive before negative, singular before plural, […] nominal before other parts of speech, count before mass, and the food hierarchy (which they give as fish > meat > drink > fruit > vegetables > baked goods > dairy products > spices).

Where does the singular countable noun arrow fit here? Apparently it defies the twenty principles listed, bow being singular, should be followed by the plural noun arrows. However, the order of the two nouns do comply with Cooper and Ross's semantic and phonological constraints: bow is the source of power hence it earns first position; it is monosyllabic and therefore must precede arrow which is bisyllabic.

Moreover, the/a bow and arrow, along with its timeless image, have both become symbolic and iconic. We associate the phrase and image with the sport, archery; military warfare; hunting; mythology and love.

cupid
2. A representation of Cupid as a naked cherubic boy usually having wings and holding a bow and arrow, used as a symbol of love.

2008, Basic Illustrated Archery [emphasis mine]

Primitive man first made use of the bow and arrow as a weapon... as archery continued to evolve and develop, more and more civilizations began using the bow and arrow to defend themselves...

Imagine for a moment someone with a bow and arrow. ... Robin Hood, the notorious outlaw and accomplished archer, is one of the most popular choices.

Pre-25,000 B.C.: early man may have invented the bow and arrow in Africa,... where the first stone arrow-heads were discovered.

British English corpus

And finally, looking at Google Ngram, we see that the expressions: “a bow and arrows” (blue line) and “his bow and arrows” (red line) were more common in the 18th and 19th century than its modern day counterpart, “a bow and arrow” (green line)

If we compare the bare binomials bow and arrow (blue line) with bow and arrows (red line) we see that the latter form is still used, but its marked descent began in the 1920s. Ngram

Citations of “bow and arrows”

1799, The New Robinson Crusoe… By Joachim Heinrich Campe

ƒo that, a little before night, about ƒix o'clock, he exerciƒed himƒelf in the ufe of his bow and arrows, in order that he might be in a condition to defend himƒelf, ƒhould he be attacked by any human ƒavage, or fierce animal. In a little time he acquired ƒuch a dexterous uƒe of his bow and arrows, that he could hit a mark, not bigger than a crown-piece at a great diƒtance

1835, Gentleman's Magazine, and Historical Chronicle

East Smithfield.
Thomas de Meose held a messuage, and one water mill, and eight acres of meadow, with the appurtenances, in East Smithfield, London, by the service of finding for the king a footman, with a bow and arrows, for forty days at his own charge, in the Tower of London, in the time of war.

1853, English Forests and Forest Trees,…

By the 6th of Henry VIII. all male servants were to provide themselves with one bow and four arrows, which their master was to pay for, stopping the purchase-money out of their wages. Another statute passed in the same reign enjoined the use of archery more extensively. It ordained that every man under sixty, except spiritual men, justices, &c., should use shooting with the long-bow, and have a bow and arrows continually in his house; that he should provide bows and arrows for his servants and children.

1855, The New Church Repository and Monthly Review

As war in the Word signifies spiritual war, therefore warlike arms, such as the sword, spear, shield, target, bow and arrows, signify such things as are proper thereto.— A. R. 299

1928, My People, the Sioux; Page 9

About the first gift I received from my father was a bow and arrows. He made them himself, painting the bow red, which signified that he had been wounded in battle.

1941, An Apache Life-Way

Conspicuous, too, are weapons of war and chase—the bow and arrows, quiver, bow cover, shield […]
As soon as the boy is provided with a bow and arrows he spends a great deal of time gaining accuracy in handling them. […]
When the boy is sufficiently strong to handle a bow and arrows, some member of the extended family — the father, the mother's brother, or the grandfather — provides him with them and gives him the necessary advice.

1954, Boys' Life, page 51

Archery can give you some of this summer's best fun, if you just know how to get the most out of your bow and arrows. You'll need bow, arrows, arm and finger guards, and — preferably — a quiver.