It is nothing more than a symbol that distinguishes one person from another. The essence of the master's name for his apprentice is the same as that of his adopted son. There are many good wishes and hopes from Master to his disciples. Different sects will be named after a so-called word generation. Some of them will be named after a recitation or cycle in Buddhist scriptures. If family members accept disciples at home, some monks will simply name a word in the middle of their names with Buddhist scriptures or the words of their disciples. For example, if your name is Zhang Ai-ren, the converted monk will change your converted name to Zhang Hongren, which will save trouble.
In Chinese mainland, 90% of the Buddhist monks in the Han Dynasty belonged to Lin Ji, because there is a saying that "the descendants of Lin Ji are all over the world".