In the old days before most of us dwelled at least part-time in cyberspace, if two similar essays were submitted to the same college, they would probably come from the same--or neighboring--high schools. While I can't speak for every admission official, my best guess is that most would begin by wondering how stupid a student could be to copy the work of a competitor applicant. However, the admission folks would not be able to tell which kid was the guilty (and stupid) one, so they'd then contact the students' guidance counselor (or counselors) to ask about the individuals involved. In some instances, the counselor might be able to shed light on the situation right away ("Leon has never had an original thought in his life") but, in most cases, the counselor would meet with the students and wait for one of them to confess (or at least to appear guilty beyond a shadow of a doubt). I suppose there could be some occasions where there is even mutual guilt. ("I'll write your essay for fifty bucks. In fact, I already have a good one in my files.")
Now, of course, thanks to the Internet, it's possible for strangers who live thousands of miles apart to share (or steal) all sorts of information, including college essays. Even so, admission officials who spot potential plagiarism would mostly likely still go straight to the school counselors with their concerns, regardless of the distance that separates them.
Occasionally, similar essays may be truly coincidental. If, for example, Brandon and Brendon are both starters on the same state-championship basketball team and both are also fairly simplistic writers, I can imagine that their "Winning the Big Game" essays might sound strikingly alike. So the only crime committed would be boring admission folks to death. ;-) Again, that's something that a school counselor should be able to clarify.
I hope you know that college administrators take plagiarism very, very seriously, and this begins even before students are admitted. So I also hope that your question is a theoretical one and doesn't spring from any actual occurrence.