Can future record holders see my ID?

If you don’t have a verified account, all they will see is a blockchain address. If you do have a verified account, it depends on what information you have made transparent in your particular wallet.

Does a Codex Record prove I own something?

Codex Records are a way to verify your ownership through existing documentation, but not a representation of legal title. An asset that is registered on a Codex Record with thorough provenance documentation can help in proving ownership.