What it actually is
Melt is flow: fat softening and the protein matrix losing enough rigidity to spread. Stretch is something else entirely: long casein strands sliding past one another without breaking, which needs the strands to be aligned and still connected. A cheese can have the first without the second.
Why it matters
On a pizza the two are judged separately and by different people. A product that melts and does not pull is rejected by the consumer; a product that pulls but does not flow is rejected by the operator. Specifying only melt leaves half the requirement unstated.
What changes it
Calcium content, which cross links the casein; pH, which controls how much calcium is bound; protein content and type; moisture; and the emulsifying salts used. In an analogue system the starch and protein chosen decide it, because there is no casein to do the work.
