Also periods of religious fasting? Why does this still apply to two of the three abrahamic religions but not the other, especially since that's the 'middle' one as it were?
Christianity, especially in its earliest days was, and in some denominations still is, full of dietary restrictions. Today Lent fasting is pretty well known, but it's a small example of all the various ways that Christians expressed their faith and devotion through either abstaining from or partaking in certain food items.
I'm going to focus largely on the Middle Ages for this question, and specifically the early middle ages. Someone more versed in early modern or modern Christianity can tell us a about how things developed later on.
In the early middle ages, and I do mean the earliest middle ages, Christian anxieties about pagan religious affiliation had consequences on the acceptable food that was available to Christian's. In Germanic contexts this often took the form of prohibitions on horse consumption.
I've laid out why this was in my answer here
During the conversion process to Christianity, among the practices that were outlawed or at least very harshly condemned, was the practice of horse consumption. Robin Fleming's Britain After Rome mentions this in the context of Anglo-Saxon England, but it was also present in Scandinavia and on the continent. The relationship between Germanic pagan religious life and horse meat did not escape the notice of Christian authorities. As a part of St. Boniface's mission in Germany, he was instructed by the Papacy to stamp out the practice of horse consumption as a pagan and abominable practice that had no place in Christian society.
Kristopher Poole makes the argument that horse meat was consumed almost exclusively in pagan sacrifices and this association between paganism and horse meat was too strong for contemporary Christians to look the other way on. He examined the relative levels of horse remains that are believed to have been consumed in England in three time periods. The levels of horse remains in each time period is rare, but much more pronounced in the Early period of Anglo-Saxon England (c500-700), pre-conversion. It then collapses quite dramatically in the middle period (c700-900), and finally concludes with a slight uptick in the Late Anglo-Saxon period (900-1100), however it does not become nearly as well represented it was in the earlier sites. The connection between horse meat and pagan sacrifice is further born out in other contexts as well such as Iceland.
These are the relevant sections.
Dietary restrictions were also common well after conversion though in both "secular" as well as monastic contexts. I'll leave discussions of expressions of faith as expressed through food to others more capable, but in monastic movements, dietary restrictions were quite common for a variety of reasons. Some of these were medical, or what Medieval medicine operated on, such as the assumption that certain foods would imbalance the humors and cause an excess of carnal desire, certainly to be avoided in monastic communities. There are other examples as well, in Anglo-Saxon England, or at least in the context of Aelfric's collory, for example, monks were expected to abstain from meat and wine was held as a more "advanced" drink for more elderly and wise men, the speaker in the text by contrast drinks ale or water.
Now these restrictions were not immutable, and indeed dietary restrictions such as fasting during Lent, are often excused to allow for sickness, childbearing, and other needs. I do hope someone more familiar with later medieval expressions of food and religiosity pops in though to detail later practices.