The following is copied verbatim from a JLPT prep book:
将棋歴30年のベテランに、初心者の私が叶うべくもない
I read it as,
A beginner as myself should be no match for a shougi veteran with 30 years of expertise.
But I would have expected かなう to be written as 敵う. Do I understand the sentence right or am I mistaken? Why かなう is written as 叶う?
I agree that this かなう is almost always written as 敵う in kanji in modern Japanese. According to 大辞林:
④（多く「敵う」と書く）対抗できるほどである。匹敵する。打ち消し表現を伴って用いる。 「二人でかかっても－・う相手ではない」 〔「かなえる」に対する自動詞〕 → かなわない ・ かなわぬ
However 適う, 叶う, and 敵う share the same base meaning ("to match"), and the kanji choice was differentiated and standardized relatively recently. So I would say it's very close to an error by today's standards, but it's not entirely an error. I somehow feel 叶 most retains the base meaning of かなう (also see the example in the link), so I wouldn't be too surprised if some novelists intentionally chose to use 叶う in this case.
Let's talk about 同訓異字： words with the same kun reading but different kanji orthographies.
As an advanced student, you undoubtedly have learned that the choice of kanji can select different nuances of the same word. It's important to choose the proper kanji for each context, right? But look at @naruto's quote of the Daijirin:
（多く「敵う」と書く）
Indeed, if we consult such an important reference as the Kōjien, look at how it formats its entries:
かな・う【適う・叶う】カナフ
① ちょうどよくあう。[…]
② 望みどおりになる。[…]
③ (「敵う」とも書く) 匹敵する。及ぶ。[…]
Are you getting the implication here? The header unifies 適う and 叶う under the same entry. What's more, sense ③ is shared with 敵う; and furthermore, there's no separate entry for 敵う. This is the only place you'll find 敵う in the friggin' Kōjien. To put it in another way, Kōjien does not think of words this way:
Instead, it thinks of words this way:
Importantly, the top-level classifier is the kana reading, not the kanji orthography. Notice that, according to the Kōjien, all of かなう／叶う／適う can be used for all three senses in any combination; while 敵う alone is restricted, and only used for sense ③.
There's a difference between what the dictionary says, and how people actually use kanji. Is it the case that 叶う is used in senses ①, ③ (to be suited; to rival) in modern Japanese?
An easy source of modern Japanese is twitter:
• 弱さで俺に叶う奴はいない！
• あの存在感に叶う奴はそうはいない
• #arslan おまえらに叶う相手じゃないよ(｀・ω・´)
• ほのかの笑顔に叶う奴はいない
• お前が叶う相手じゃ無いんだから
There's plenty more, so 叶う in senses ①, ③ is quite alive. Judging from a few more searches, 敵う is almost always sense ③ (to rival), and 叶う is usually sense ② (to come true), but not necessarily. Kōjien is correct, even for bleeding-edge Japanese.
Let's think about this phenomenon from another angle. The English language, as you know, doesn't distinguish oyu from mizu. Somehow they get by by calling both "water" (when the distinction is needed, they might qualify it as "hot water" or "cold", but often it's just unspecified). Imagine translating an English cookbook to Japanese: for every occurrence of the word "water", you have to decide whether it's oyu or mizu, and select the adequate translation in context.
For a long time, the Japanese wrote in Chinese (kanbun). Indeed, they wrote in Chinese before even writing in Japanese. This fact had a huge impact in what Japanese writing is like. Originally you have only one Japanese verb, kanau, with the three different senses. But, in early Japan, if you wanted to write this word down, you'd have to translate it to Chinese kanbun first. And, like "water", there are several translations to choose. 敵 dí is "to rival", sense ③; 適 shì is "to be appropriate" ①, and also works for "to arrive at" → "to come true" ②. (As for 叶, its original meaning is yè/xié "to harmonize, to unite"; I think its Japanese uses are later extensions.)
Eventually they started using kanji to write Japanese, not just Chinese. By reversing the translation process, you get an orthographic rule: when writing kanau, use 敵 if you mean "to rival", or 適 if you mean "to be suited for; to arrive at". So a word that originally has many meanings may have each meaning clarified by the choice of kanji (cf. あう → 会う、合う、遭う etc.).
This is a pretty neat trick, allowing greater clarity in writing. However, you should not lose sight of the fact that, in speech as well as historically, these are still merely nuances of the same word; because the orthographic kanji distinctions won't always be followed to the letter. You should still learn the dictionary recommendations so that you can write "properly" (not to mention pass the JLPT and other exams); but, at the same time, be prepared to find a lot of variance in the real world.
I agree with @chocolate's comment that says it is a serious mistake as 叶う usually means to match when it is written in kana as the Jisho link indicates:
to match (implies competition); to rival; to bear (e.g. I can't bear the heat) Usually written using kana alone, esp. 敵う, usu. with neg. verb
I think your sentence could be better translated to
A beginner as myself cannot possibly be a match for a veteran with 30 years of experience in Shogi (Japanese chess).
or
I, a beginner as myself, cannot even think about competing with a veteran with 30 years of experience in Shogi.
[歴]{れき} usually implies experience or history. It doesn't necessarily mean expertise.
"叶う（適う）" and "敵う" are pronounced same, but are very different in meaning, as you understand.
"叶う" means to match, meet, satisfy the requirement and conditions, as in "夢が叶う - the dream comes true," "条件に適った家 - the house that meets my requirement," while "敵う" means to rival, vie with, as in "奴さんはとてもお前が敵う相手ではない - He isn't a guy that you can beat."
In your quote - 初心者の私が叶うべくもない, "叶うべくもない" is mixing up the usage of "敵う" and "叶う." It's a typo, which should be written as "敵う."
