Which part of a DNA molecule carries the genetic instructions that are unique for each individual: the sugar-phosphate backbone or the nitrogen-containing bases?

the nitrogen-containing bases

Explanation:

The nitrogen-containing base sequence of DNA contains all the genetic information which are encoded as mRNA during transcription and then translated into polypeptides during the process of translation/protein synthesis.

The genetic code (codon) is a sequence of 3 bases which codes for specific amino acids.

The sugar phosphate backbone of the DNA holds the DNA structure together and has a negative charge that makes the DNA to be water-soluble.