Hey everyone! I really appreciate all the advice.
And I have an interesting update!
I received an email back from the department chair. She said that I should address the situation directly with my professor first, and if I still had concerns to talk to her then. She also made it very clear to me that the professor is not aware of which individuals (or even in which sections) made complaints about him.
So I went to my professor's office hour today and...
I learned that when he said identical, he meant identical. I actually think he's on my side in this one, but basically some other student in the class somehow got a hold of my code and submitted it without any changes whatsoever. I find that kind of ludicrous, but that's the only thing that I could conclude given that he said they were identical, even including the comments.
So, tomorrow after class I will be meeting with my instructor and this other student to talk about this. I am positive that I was the first one to contact my professor, and in person as well. We talked about different possibilities that someone might have had an opportunity to steal my code, but I really couldn't think of anything. I never leave my computer or flashdrive unattended, and I do most of my coding at home.
Worst comes to worst, I also have maybe 3-4 "prototype" files on my flashdrive and computer which I have been very careful not to open given that they are dated well before the assignment's due date and could possibly be used as proof.