The Muslims believe that you can only truly read the Koran in Arabic
The relationship between the Quran and Arabic is very different from that of the Bible and Greek and Latin. To a large extent, the Arabic language was
defined by its use as the composition language for the Quran, which is considered the stylistic and literary zenith of the language. As it were, Arabic scholarship is somewhat disinterested in pre-Islamic Arabic, and some Quranic commentators suggest that the Quran and Arabic are so inextricably linked on an essential basis that it is impossible to separate the two.
That is very different from the situation of Latin and Greek in Christian antiquity. The Greek and Latin scriptures (whether in original compositions or translations) were composed when their respective languages were already quite mature and possessed a sophisticated literary corpus. The Greek and Latin Fathers rarely, if ever, looked to the Greek or Latin scriptures as literary models that defined the language. In the case of the Greek Fathers, they often ‘Atticised’ their language (in imitation of Classical Athenian literature) and modelled much of their writing on the Alexandrian Canon (ten orators of antiquity).
Likewise, the Vulgate was often a tug-o’-war between its ‘vulgar’ origins and attempts refine its style in a more classical mode. This tendency towards ‘Classicisation’ can be seen quite ready when contrasting and comparing the gradual evolution of the Vulgate from the (pre-Jerome) Vetus Latina, to Jerome’s Vulgate, to the Sixto Vulgate, to the Sixto-Clementine Vulgate and then to the current Nova Vulgata.
In addition to what others have said, I will note that it’s very important to avoid fetishising the Vulgate and drawing tendentious connections to pre-Christian Rome: it is not the Latin that Cicero or Virgil spoke, nor is it characteristic of Republican Latin. It’s important to appreciate the Vulgate and other Latin Christian texts on their merits.