Before I answer, let me first specify that while the popular conception of illuminated manuscripts involves production at monasteries, and while this did continue throughout the Middle Ages, the age of the monastery as the primary locus for creating illuminated manuscripts is the early Middle Ages. By the 12th century manuscripts are increasingly being produced by professional, non-monastic scribes, and most of the illuminated masterpieces of the later Middle Ages are associated with secular workshops rather than monastic houses.

So let's look at the early Middle Ages. Many will be familiar with the Book of Kells, an illuminated manuscript produced in a monastery somewhere in Britain, Scotland, or Ireland around 800. Some may also know the Lindesfarne Gospels, produced at Lindesfarne (in Northumberland) around 715-720. Looking at these two books, you will notice stylistic similarities, especially in the insular majuscule script employed and the general look of the decoration, such as the use of interlacing or "knotwork" motifs. For Lindesfarne, we even know the name of the scribe-- Eadfrith. But they are not the only books done in the same area in the same general style - Lindesfarne, for example, is also responsible for the Durham Gospels; also from the same general time period are the Book of Durrow and the Gospels of St. Chad (which may have been influenced by Eadfrith's work). These monasteries definitely seemed to be in contact with each other and were influenced by each other, and although there are stylistic differences between all of them, they certainly have a similar style--which is generally termed the Insular style (named after the islands of Britain and Ireland, where it flourished).

Was there an effort, then, towards standardization and uniformity? Not per se, in that we rarely see outright copies. However, there is definitely a language of standard motifs for these Gospels, particularly in the symbology of the four Evangelists ( a man for Matthew, an eagle for John, a lion for Mark, and a calf for Luke). The texts themselves--were certainly uniform. But the fact that we know the name of at least one scribe suggests that there was considerable leeway given to artists within the general constraints of producing illuminated Gospels.

Note also that not all manuscripts were so lavishly illuminated. Monasteries did use other books as well, such as saints' lives and the like, that were usually more simply decorated, although the decorations do reflect the same general stylistic conventions as their more ornate cousins.

Outside of the British Isles, individual monasteries definitely did develop their own styles--enough so that one can often identify them by the use of a particular variant of script or illumination style. Sometimes, as for Eadfrith, we can even identify a particular scribe who becomes known for his style. The Evangelist motifs popular in the Insular style were also popular on the Continent, although they had a bit of a different look. But there was definitely no attempt by, say, the Church as a whole to dictate a particular style throughout the West. The text itself was what was important--at least for Gospels-- not the decorations, and this was certainly much more standardized and less susceptible to local variants.

Sources/further reading: Janet Backhouse's The Illuminated Manuscript; Raymond Clemens and Timothy Graham, Introduction to Manuscript Studies.