Are college students forbidden from sharing their grades with other students?

I have a lifetime of experience in academia in the US, and in all of my experience your instructor's claim is completely without merit. There is no regulation or cultural norm that requires students to keep their grades confidential from other students. Moreover, comparing assignments or exams for learning purposes and/or to confirm that the grades have been assigned fairly is a very common practice among American students: at any American institution I would assume students have a "right" to do so unless given explicit information to the contrary.

So, bottom line: there is a very good chance that what the instructor has told you is bogus. What should you do about it? I would begin by politely asking your instructor whether there is any written confidentiality agreement that you are subject to. If he says yes: good to know! If he says no: I would seek aid from someone at your institution. Good places to start are (i) your faculty advisor or (ii) your student ombudsperson. Eventually you may want to speak to the department head and/or the relevant dean, but I would proceed carefully and get as much advice as possible rather than escalate too quickly. It is very likely that you are in the right here, as others should recognize without your needing to press too hard or further antagonize your instructor. (Someone at an American institution who says this to a student is likely to be well towards the "unreasonable" end of the spectrum, so I would try not to hand them an excuse to retaliate against you.)

Good luck.