Can a tag prove it is genuine?
Only if it holds something it never discloses and can demonstrate knowing. A tag that states an identifier, a serial number or a signed message is asserting, and every assertion can be read and rewritten onto a blank. Proof requires a secret that stays inside the chip.
Asserting and proving
An assertion is a statement the tag hands over. Anybody who receives it can repeat it, because receiving it is having it.
A proof is a demonstration that the tag knows something, arranged so that watching the demonstration does not let you perform it.
Every question about tag authenticity reduces to which of those two you have, and almost every disappointment comes from buying the first while believing it is the second.
Everything that only asserts
The identifier. It is announced to any reader before anything else happens, so it is public by design.
A serial number written into the content. Readable, and therefore writable to a blank.
A signed record. The signature verifies on the copy exactly as it does on the original, because it is part of the content that was copied.
A password on the chip. It gates an operation and says nothing about which tag you are holding.
An encoding nobody understands. Obscurity is not a secret, and a copy does not need to understand what it is copying.
What proving requires
A key inside the chip that never comes out, and an exchange in which the chip answers a question in a way only that key permits.
The answer is different every time, or specific to the question asked, so that recording one exchange yields nothing reusable.
And a verifier on the other side capable of judging the answer, which is the half people forget and the half that costs money.
Which chip families provide this and what each is called belongs to the manufacturers' documentation, and the terminology differs between them.
The tag is not the item
Even a chip that proves itself perfectly proves only that this chip is genuine. It does not prove that the bag, the bottle or the part it is glued to is genuine.
A counterfeiter who obtains a genuine tag and applies it to a fake has defeated the scheme without breaking anything cryptographic.
Closing that gap means binding the tag to the object physically, with a tamper construction, and recording which tag belongs to which item.
Any scheme that treats chip authenticity as product authenticity has a hole in it, and the hole is where attacks actually go.
Frequently asked questions
Is a serial number on a tag proof of anything?
No. It can be read and written to a blank tag. Anything readable is reproducible.
What about a signed record?
The signature copies with the content and verifies on the copy. It proves authorship of the message, not identity of the tag.
What actually proves a tag is genuine?
A key inside the chip that never leaves it, demonstrated through an exchange that cannot be replayed.
Does that prove the product is genuine?
No. It proves the chip is. A genuine tag on a counterfeit item defeats the scheme without any cryptography being broken.