Was "book" to "beek" as "foot" is to "feet"?

This will be a short answer, because I cannot research the topic right now.

I don't know whether in Anglo-Saxon times book, or better its equivalent, might have had a plural form similar to the one foot has nowadays. What I know from reading Chaucer's "Canterbury Tales" (14th century English) is that at that time the word was spelt "bookes" in the plural, as you can read in the passage about the Oxford Cleric :

For hym was levere have at his beddes heed
Twenty bookes, clad in blak or reed,
Of Aristotle and his philosophie ...

Besides, the plural form in double 'e' is surely not a one-off as you suggest; the same is true for tooth - teeth and goose - geese (and perhaps something else which does not come to my mind right now).

Finally, we should remember that there are words which are written with two o's but the corresponding verb is written with two e's (for example, blood and to bleed)

Here is a quote from the Old English dictionary:

bóc [] f (béc/béc) 1. a book, a document, register, catalog; 1a. a legal document, (1) a bill of divorce; (2) a charter; (3) a title deed; (4) conveyance; 2. a book, volume, literary work, pages; main division of a work;

Let's now look at:

fót [] m (-es/fét) 1. a foot; 2. the foot, the foot of a man

So, I guess yes, we can say that at least the word from which modern book evolved had plural in the same form as the word from which modern 'foot' evolved. I guess, as early of Middle English period, "book" lost its irregular plural, since Chaucer used bookes.

It's also interesting to notice that fót actually had an alternative form fótes, but this form has not survived.

According to the Online Etymology Dictionary this is an instance of I-Mutation

I-MUTATION (also known as "i-umlaut") is the raising and fronting of a root vowel in anticipation of "i" or "y" sound in a suffix.

Other examples of noun plurals that exhibit I-Mutation are

man → men
tooth → teeth
goose → geese
louse → lice
mouse → mice

I-mutation is caused by the very human habit of laziness: taking the shortest distance between two points. The plural of man in ancient West Germanic, the ancestor of Old English, used to be a word something like *manniz. The speakers "cheated" on the first vowel in the word to be in position for the second vowel. It's the same thing you do with doing. It doesn't change the meaning of the word to do so.

So, it comes from the old Saxon roots of English. Book, being a relatively new and uncommon noun compared to foot, probably wasn't shortened so commonly.

As a rule, morphological irregularities of Modern English are remnants of previously regular processes of Old English.

So, if English had been a more conservative language, we would have retained beech as the plural of book. Moreover, we would have had book as the past tense of bake (like took and shook from take and shake).

German, which is more morphologically conservative than English, still has vowel alternation in both these places. The plural of Buch “book” is Bücher, and the past tense of back- “bake” is buk (though a regular form, backte, is emerging, just like English baked).

The plural of foot always was feet; we inherited the word from West Germanic, which seems to have had a lot more plurals where the vowel changed than English does today. The word for "foot" in German is Fuß (rhyming with "moose"); the plural is Füße. You see the umlaut: that means the vowel changes, just like it does in English. The word for book in German is Buch; the plural is Bücher, so you see the vowel changes in that as well. Somewhere along the way, English lost the irregular plural of book, and most of the other vowel-changing plurals as well. Only a handful in very common words survived.

In the old languages, such as Old English and Old Norse, which are the largest sources for modern english, book was derived from Bok. As foot was from fot. Multiples were boks and fots. By speaking these words in a fast and lazy manner, as people do in regular conversation, the words eventually evolved into Beeks and beek for the plural book, and feets and feet for the plural foot. During the transition of the 13th century, the english language started becoming more organized. English kings were all speaking old french, but demanded a unification of the English language for better unity of the people to be ruled. By the end of the 13th century, beek was no longer common and had been plurilized by an "s", thus we now use books. Feet remained in common language because everyone who spoke english mention foot and feet on a near daily basis. Books were not so common to the average englishman. The words that kept the oral mutation, such as foot and feet, woman and women, mouse and mice, etc, were all words used on a daily basis by the common man of the time.

Old English bōc did indeed become bēċ by umlaut. I don't know when the -s form rose, whether it was already present in OE or not (bōcas?), but that form eventually won out. Other words which lost their umlauts include oak (OE āc/ǣċ) and nut (OE hnutu/hnyte).