If one professor gives recommendation letter to too many students then does it affect the admission of students?
I think that is pretty unlikely, as long as the professor is honest in what they say. I'm assuming, of course, that these letters go to various places, not all the same. But, if everyone is applying to the same university and gets the same letter, few will succeed. But that is at least partly because the number of open slots is limited. 

I think this should not be a worry unless the situation is very specialized.