Does diamond has serial numbers?

Yes.  Most diamonds that are accompanied with major Diamond Labs Certificates usually have the serial numbers engraved into the girdle rim of the diamond.

This ensures that that very specific diamond belongs to that specific certificate.

Diamonds with laser inscribed serial numbers tend to value more in terms of price.

Simply because customers who do not know much about diamond can feel secure when purchasing a diamond with serial number inscribed and accompanied by the diamond's certificate.

It is also important when trying to resell the diamond as you can show the certificate to potential buyers assuring them that the diamond is Authentic and have certain characteristics as stated on the diamond's certificate.

The serial numbers on the diamond is not visible to the naked eye, you have to use a 10 x magnifying glass (scope) to be able to match the serial numbers on the diamond with that on the certificate.

You can go to any diamonds jeweller and tell them you want to verify the serial numbers. Most of them offer this service free of charge especially if you bought the diamond from their store.