A professor requested that I exchange my paper with another student, to get their suggestions. Every student in the course must exchange their papers in this way, before the teacher will look at it. I have never encountered another professor with such a policy and am concerned of the possibility that a student might claim my paper to be their own. What steps can I take to guard against such a possibility?
Email the work to someone (or yourself) well before you hand it over.
Also, try to relax. It is not likely someone will try to pass your work off as theirs because they would be aware that you were going to hand the same piece of writing in and that would lead to trouble for both of you.
I think the reason he is getting you to do it is to get you used to how things work in a research environment. So it is not such a bad thing, maybe.
This seems like a reasonable request by the instructor. Without understanding why you are feeling vulnerable, it is not possible to provide an answer. It is worth talking to someone (possibly the instructor or maybe your advisor or a therapist) about why you are feeling vulnerable.
As for protecting yourself, emailing the document to/from your university email account is probably sufficient as documentation. The next level up would be to email the student the document with the instructor cc'd. This way you are proving you shared your paper (and gaining documentation). Anything more and you will likely be revealing your insecurities, which might lead to more insecurities.
Sign and date a copy of your work, seal it in an envelope (using tamper-evident seals) and have it received by your teacher (that is, give it to your teacher then have your teacher sign a note that he/she received the envelope). This will serve as evidence in case another student claims your work is his/hers.