For example, here's the in-game text that pops up when Theravada Buddhism is "harmonized":
The Pali Cannons of Theravada Buddhism are in many ways a familiar way to organize tradition and learning to our Confucian elites. Over time these collections of laws and rites have come to be studied and made part of the corpus that every educated and influential member of our society must study and adhere to. The tradition have been harmonized and the wider Theravada Buddhist community consider us their equal just as we consider the Theravada Buddhists within our borders as an integral part of our monarchy.
and when Islam is "harmonized":
Islam has been a minority religion for centuries in our lands and was so not a new religion to us when we first became the overlord of the people of Kutai. Over time however it has grown into a more and more integrated part of our society. Muslim ideals have been integrated and harmonized with our traditions, rites and methods and we have taken the best of what the religion has had to offer to mould to our own uses. Islamic scholars, doctors, merchants, farmers and peasants can be found even in the districts around Beijing. As long as they bend the knee to our king, the followers of Allah are an integral part of our society.
I know various belief systems such as Benzhuism or Manichaeism (as it entered China) became influenced by Chinese folk religion, but I believe that has more to do with being influenced by their location, rather than any state-supported attempts to integrate them.
With that being said, was there any precedent behind a Confucian ruler trying to officially integrate these religions in such a manner as to incorporate their rituals, traditions, and even religious texts?
While the nomenclature seems to be a convenience for the game mechanic, the Chinese have long been of the "buffet" mentality (take what you want, and leave the rest), even in official contexts.
To understand the role Confucianism played in the ruling classes in Chinese antiquity, it's necessary to separate Confucianism the "religion" and Confucianism the "operating philosophy", of which the former is a ritualised and minority part. To put it as simply as possible, Confucius was a philosopher who laid out the foundation upon which most East Asian societies would go on to be built, whether through imperialism, vassal statehood, or cultural exchange. It is especially notable for its prescriptions concerning filial piety, class hierarchy, and the proper etiquette between and inside those hierarchies. It serves as the blueprint for the whole of society, and was something of a default presumption for any living within that society.
Of particular note here is that Confucian thought, though not religious in the strictest sense, is predicated upon certain presumptions concerning the super- and supranatural inherited from Chinese folk religion dating back to antiquity; while it does not make any great effort to detail the heavens by the grace of which society is allowed to function, it does accept them as a matter of fact. This presumption manifests itself also in the ruling classes, all the way up to the emperor (the ideogram for whom depicts him as the literal tether of heaven and earth) and the so-called "Mandate of Heaven" via which the dynastic cycle — in each rise and in each fall — manifested itself for the benefit of all man.
This is all to say that the Chinese (and especially in matters of state) were not religious, per se, but they were by no means strangers to what some may see as religious ideas or thought. Historically, a Chinese was simultaneously Confucian, Daoist, a practitioner of Feng Shui, an observer of the zodiac, and all but for a single, ancient presumption of the auspices of heaven such that even the decidedly areligious court were, by virtue of being allowed to govern the earth at all, complicit members of the so-called Chinese composite religion, even if the rulers themselves were foreigners; those who were not apt to buy into China's ways were not long to rule her.
This, however, is in no way to say that China's mind could never be changed. Rather, this basic framework was so casual as to allow for much more flexibility of influence than other similarly "divine rite"-style hierarchies built on more stringent definitions of what divinity could mean. Thus, it was historically very easy for the Chinese to reconcile the tenets of many outside religions with their own, and very casually at that, as the framework into which these tenets were being adopted did not, by and large, exist definitionally, and thus not by some necessary distinction from another. For the Chinese, "religion" was by default an inclusive, simultaneous affair, not a mutually exclusive one. In a certain manner of speaking, the facts of life were not necessarily being changed, just amended.
This remained true for most of China's history because the threat of her downfall was never of any real concern. When that changed, we would see very different things happen. But methinks that is the subject of another question.