I have hard time understanding why in this phrase, property is preceded with a definite article. As this book is one of many properties, the indefinite article sounds more natural, to me.
Of course, I'm wrong, but I would like to know why.
I don't think that either of the previous answers truly nails the rationale behind this usage. I don't think the following does either, but I consider the reasoning a useful addition.
The bank has a responsibility to inform customers of changes in interest rates.
The bank has the responsibility to inform customers of changes in interest rates.
are quite acceptable.
This argues that qualification of a noun is not the sole factor involved.
(Though these versions are near-interchangeable, the one with the definite article carries at least a hint that the need to inform people of interest-rate changes is common knowledge. And, although the version with the indefinite article may be said to carry a hint of other responsibilities, I'd say that this isn't a count-noun usage.)
Perhaps it's possible only to say that the definite article is often the default choice (or perhaps on occasion just a choice) with non-count nouns in expressions that have become idiomatic:
[This accident was] the fault of ... [nobody in particular, say]
the responsibility of
the property of
[Did they have] the chance of
He had the nerve to ...
felt a/the need to
... but note
have a/n duty / obligation / mind to.
The question on whether to use a definite article or indefinite article is not easy to answer. (Please note I used "the" instead of "a" in this sentence).
In your example "This book is the property of...", "property" is not any other property, but the property modified (restricted) by the prepositional phrase that is headed by "of".
The following Merriam Webster definition 2 b (1) explins that "the" is
used as a function word before a noun to limit its application to that specified by a succeeding element in the sentence: 'the poet Wordsworth", 'the days of our youth', 'didn't have the time to write'
As you can see, the nouns in the three examples are all modified – by a proper noun Wordsworth, prepositional phrase of our youth and to infinitive to write respectively.
[This answer relates specifically to British English.]
Property (in the sense of possessions) is a non-count / uncountable / mass noun.
Oxford Dictionaries define it as follows:
A thing or things belonging to someone; possessions collectively
she wanted Oliver and his property out of her flat
the stolen property was not recovered
When arrests were made stolen property was recovered that belonged to two victims of a robbery in the town centre.
He also faces a charge of possessing stolen property in connection with the March 11 incident.
Police are recommending charges of possessing stolen property, and break, enter and theft against the man.
Law The right to the possession, use, or disposal of something; ownership:
rights of property
But how were those notions of ownership and property understood in customary terms?
Just because of the abuse of property ownership, private property should not altogether be eliminated.
The second major element in communist doctrine was the social ownership of property and central planning of the economy.
Cambridge Dictionaries define it as follows:
an object or objects that belong to someone
The club does not accept responsibility for loss of or damage to club members' personal property.
Both books have "Government property" stamped inside them.
Children need to be taught to have respect for other people's property.
specialized: law the legal right to own and use something
The new tax system would be calculated on the value of property owned by an individual.
Apparently he was sacked after he was caught stealing company property.
The police impounded cars and other personal property belonging to the drug dealers.
The fire resulted in damage to their property.
In an auction, goods or property are sold to the highest bidder.
Because (in the senses mentioned above) property is a mass noun, it is not normally used with the indefinite article "a", nor in the plural form.
The expression quoted in the question: This book is the property of ... therefore uses "the property".
Likewise, one could say: "This book is my property."
For the same reasons, one could not say: "This book is one of my properties" or "This book is a property of mine". The use of the word property in the plural, or as a singular count-noun with the indefinite article "a", in that way would immediately make one think of land or buildings, rather than possessions (for the reasons given in the next paragraph).
Note that, when used to refer to a building and/or land, property can be used either as a mass noun or as a count noun, and therefore can also be used in the plural form, or in the singular form with the indefinite article "a". (See the dictionary references cited above for further details.)
Additionally, when used to refer to "an attribute, quality, or characteristic of something", property is used as a count noun, and therefore can also be used in the plural form. (See the dictionary references cited above for further details.)