"Foot" is a curious word in English because it is pluralized in an unusual way; the "oo" in the word is changed to "ee". Did this once use to be a standard way of pluralizing things in English (or a language that contributed to English), which would mean that the plural of "book" was "beek" instead of "books"? Or, is "feet" just a one-off?
Whenever you find an O (or some other back vowel like A or U) in one form of an English word and an E (or some other front vowel like Æ or I) in the corresponding place in another, you have two suspects to interrogate.
If the two words are not from the same language, but from two separate Indo-European languages, like Latin and Greek (e.g, ped-al from Latin and pod-iatrist from Greek, both roots meaning 'foot' — the p ~ f and d ~ t alternations are Grimm's Law in action), then what you're seeing is E–O Ablaut. Proto-Indo-European often alternated between an "E-grade" and an "O-Grade" form for morphology, and various daughter languages inherited various words. Sort of like the distribution of family furniture when the parents die. That's not what happened here, however.
If the two words are from English, and not borrowed, then what you're seeing is Umlaut. This term refers to changing a back rounded vowel [o, u] to the corresponding front rounded vowel [œ, y] in anticipation of a front vowel [i, e] in the next syllable. This is a common Germanic feature, and is still productive in German, where there are special vowel symbols (ü, ö, ä, called U-umlaut, O-umlaut, and A-umlaut) that represent this phenomenon, and these sounds. Umlaut is the guilty suspect.
In English, this happened to many normal plurals because of the E in the regular plural suffix -es. That E was pronounced in Old English and Middle English, but not in Modern English; however the root vowel had been changed already and is maintained in some, but not most, of the nouns.
The original Old English (or possibly Proto-, West, or Low Germanic) of goose/geese (in Modern English [gus/gis]) was [go:s/go:ses]. There were several steps in the derivation:
- dropping final s ⇒ [go:s/go:se]
- fronting the o: to œ: by umlaut ⇒ [go:s/gœ:se]
- derounding œ to e ⇒ [go:s/ge:se]
- final e going silent ⇒ [go:s/ge:s]
- Great Vowel Shift raising all long vowels one notch ⇒ [gu:s/gi:s]
- losing long vowels ⇒ [gus/gis]
Edit: In this medium, where writing and typography has to express speech and sounds, I use italics and boldface like this:
- I use plain italics only for citing examples and titles. Never for emphasis.
- I use boldface for emphasis. These are words that would be LOUD in my speech.
- I use bold italics for technical terms, usually with capitals, and links if I have them.
- I also use bold italics in examples to point out individual parts that get mentioned in the text.
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
It's also interesting to notice that fót actually had an alternative form fótes, but this form has not survived.
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)
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.
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.
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).
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.