> (Side note: "identification number" really means "opaque string". IDs are not numbers. If adding or multiplying them together doesn't make sense, they're not numbers.)

Eeehhh I'm not sure if this necessarily true. A telephone number is definitely a number but addition doesn't make all too much sense on them.

If you want to specifically talk about objects that require addition and multiplication to be properly defined on them then you can always use Groups and Rings.