The context of the comment, combined with my own intuition, makes me think this comment was referring mainly to vocabulary. As a random example of a Latin word that is not notably "precise" compared to modern European languages, consider mūs, which can refer to both mice and rats, as well as (when qualified) some other animals. The genus Mus has a very specific definition, but this is a creation of modern scientists, not an inherent property of the Latin word that is the source of the genus's name.
I believe the overall number of words attested in "Classical Latin" is not enormous compared to other languages; and I have also encountered an idea that the attitude of Classical Latin authors towards the formation of new Latin words was less permissive than in other languages (such as Greek).
Here is a passage from Frederic Taber Cooper's Word formation in the Roman Sermo Plebeius (1895) that gives this explanation (although I am skeptical of a number of aspects of Cooper's account of the nature of "sermo plebeius", a controversial topic, it seems to me that the data is valuable in any case):
In the prisca Latinitas the vocabulary was a somewhat limited one, as was natural among a people whose time was largely divided between simple pastoral pursuits and local warfare. They inherited, however, from the common Indo-Germanic stock abundant facilities for forming new derivatives and compounds at pleasure. When the schism arose between the classic and plebeian speech, the latter naturally retained these facilities, and, if we may take Plautus as a criterion, availed itself of them with characteristic license. The literary language here presents a marked contrast [...] under the formalizing influence of classicism, Word-Formation, in common with all other linguistic growth, came to a standstill, at the very time when Rome was most in need of a wider vocabulary. The consequent inferiority which Latin, in this respect, shows to Greek has been frequently acknowledged by Roman writers; Gellius dwells at length upon the difficulty of properly rendering Greek compound words, either by a single word or a periphrasis, while his citation of the diverse attempts of the early grammarians to render προσῳδίαι, by notae uocum, moderamenta, accentiunculae, uoculationes, aptly illustrates the inaptitude of the language for technical expressions. Lucretius, Cicero, and Seneca, in turn complain of the lack of a philosophic terminology: yet philosophy is but one instance of the many avenues of Greek thought opened to the Roman mind only by deliberate coinage of the necessary vocabulary. The same difficulty confronted the medical writers; Celsus expressly deplores the superiority of the Greek language, while both he and his successors adopt the expedient of introducing Greek medical terms in their writings, usually accompanied by an attempt at translation or paraphrase. The ecclesiastical writers were still more hampered, owing to the wider gulf which separated their teachings from the daily life and thought of classic Rome. In spite of the industry of Tertullian, who is rightly regarded as the creator of ecclesiastical Latin, his successors, like Hieronymus, often felt the poverty of the language, in contrast with the richness of the Greek and Hebrew, which they were striving to interpret.
There were, as Cicero himself has pointed out, three ways in which the deficiencies of the vocabulary could be supplied; either by the transfer of a Greek word bodily into the Latin, by the use of an existing Latin word in a new sense, or by the formation of a new word. But in the classic period the use of foreign words was felt to be contrary to good taste and was accordingly avoided as far as possible, while unusual expressions, either archaisms or neologisms, were severely discountenanced. Even Cicero, who did more than anyone else toward giving currency to new formations, introduced many excellent and sorely needed words with hesitation and apology. Quintilian, while admitting that new words must occasionally be risked, says frankly that even when received into the language they brought little credit to their author, and if rejected led only to ridicule; and Gellius, still more emphatic, declares that new and unknown words are worse than vulgarisms.
This extreme attitude, however, had become untenable long before the time of Gellius; a point had been reached where growth of vocabulary was essential to the life of the language. But it was a natural consequence of such conservatism that no process existed for forming a literary vocabulary possessing distinctive features which might stamp it as a cultured product; no scientific nomenclature corresponding to the -ologies, -isms, and -anas of our own language; there was not a single suffix which could be regarded as distinctly classic, and which was not comparatively more abundant in authors of inferior Latinity.
(xxxi-xxxiv)
I can't think of any good objective measurement of the "imprecision" of Latin speech. You could look at vocabulary size, however, that is a rough proxy and is difficult to measure. Kingshorsey's answer to this previous Q&A indicates that "The Oxford Latin Dictionary has about 40,000 entries". There is some discussion on other measures of Latin and Greek vocabulary size here: vocabulary size of classical Latin and Attic Greek (Textkit)
