How does blockchain help here? The database would be extremely large - too large to hold for any ordinary computer system.
The basic problem is solveable without blockchain by simply registering products by their serial number on the brand's website, which a lot of brands actually already do for higher priced products.
You don't need to link it to any customer details. The way it usually works is the website records every serial number checked, and warns you if somebody has already checked that serial number. For additional assurance the serial number can also be concealed with tamper-evident packaging, e.g. scratch-off paint.
D’Addario has a “serial number” check webpage for their guitar strings. If you enter an invalid (or previously used) serial number, supposedly (I’ve never had a counterfeit set) they’ll send you a genuine set for free.
Sure, some people will input fake serial numbers to get a free product, but they probably weighed the cost of counterfeit strings affecting brand image (if I don’t know mine are counterfeit, I’ll assume it’s D’Addario’s fault if they’re bad) with the cost of sending a free set of $5 strings.
The basic problem is solveable without blockchain by simply registering products by their serial number on the brand's website, which a lot of brands actually already do for higher priced products.