A dove house or dove cote is a building where you keep doves or pigeons for meat, eggs, manure and so on. As homing pigeons became more commonly used they might also be housed in a cote structure but in medieval Christian Europe they were primarily livestock. Dove eggs, dove in sauce, dove dishes of every type were prized by nobles; especially in the winter when alternative meat was scarce.

In many regions only noble families could keep dove cotes. Cotes could be found on manor grounds and trained keepers were a part of a noble’s staff. In France it was known as le droit de colombier— the right of the dove (http://rvbrian.over-blog.org/article-24253379.html). Why the privilege was limited to nobility (or as the law you cite hints, those who have been raising birds for a long time) is up for debate. One site (http://www.pigeoncote.com/dovecote/dovecote.html) suggests it was a means of social control, likely similar to sumptuary laws or poaching penalties. The lower classes could not be seen to be imitating nobility, nor could they gain too much independence. Keeping doves was a privilege with benefits and restricting it kept the social hierarchy in place.

There might have also been some religious/symbolic elements— doves are mentioned in the Bible and associated with chastity and the soul (https://www.traditioninaction.org/religious/f037_Dove.htm). So in addition to being kept by noble households they were also a part of monastic life. Monasteries and nunneries often kept doves or pigeons.

In England there were some 26,000 pigeon cotes in castle and monastery grounds. Most followed the same general layout— they were often round with small nesting alcoves in the walls and a ladder to reach the nests— but the details varied. Since they were status symbols architects strove to outdo each other (https://slate.com/human-interest/2015/09/the-history-of-the-dovecote-is-all-about-birds-and-status.html). Although dove cotes began as simple structures, as useful as a stable, in later years they became more charming garden features (https://www.buildinghistory.org/buildings/dovecotes.shtml). Many can still be found today in historic buildings.