Given 0s on Assignments with suspected and dismissed cheating?

It seems that you are being treated unfairly, but it is a situation that can only be judged and handled locally. Your university probably has appeal processes and you can always go to the department head.

In programming as in mathematics there is often only one clear way to do something and if different students do "the expected" thing, then their programs come out similar - occasionally very similar.

"Suspicion" of cheating should never be the final determinant.

But no one here can help you. Seek a solution locally.

I have heard about professors threatening all kinds of stuff because of alleged cheating. In our university, such cases may not be judged by the professors or teachers of the course (and doing so, especially making threats can get them into trouble). Instead, they are required to present it to a commission consisting of a few higher-ups in the department.

I suggest you look up how this works at your school or university. I imagine you can ask your mentor or adviser (or just another professor / teacher you trust) what the procedure is there.

It shouldn't be a problem to ask what the procedure is, and if you like what you hear you can always try to make your case. Though I'd informally explain the situation to someone you trust, e.g. mentor, adviser, etc. first. If you are willing to make a case, you can probably tell the teacher beforehand, if they don't think they have a case they might drop it all together because they don't want involve others (especially if they have little proof).

Once you do involve other people (teachers / some commission), make sure to make your case as tight as possible. If you can prove that you wrote it (e.g. a cloud service that shows when you saved, Github commits or chat logs showing your worked on it together) then that can help you convince those who are judging the case.

You say you've already talked to your professor 1-on-1 and showed that you indeed coded the assignment yourself. If so, I suggest getting your professor to also talk to your friend 1-on-1. If your friend can also show that he coded the assignment himself, it would be sensible to give both of you the points. If your friend can't, then you have a good case for why he should be getting the zeros, and not you.