If you fly into country C, yes the airline may disclose what passport you used. So yes country C may technically know about both. Also it’s legitimately possible for 2 people to be born on the same day with the same name. They simply don’t care.

You won’t get any official documentation anywhere about any of this, due to the nature of it. You can find heaps of evidence of it working fine by reading up forums on the matter.

Are you a dual national with at least one of the countries allowing dual citizenship? Or do both prohibit it?