Why does adding commercial processed cheese to fondue change its consistency?
The magic is from Sodium Citrate

Most mass produced cheese it based on "cheddar blends". Basically large (50 Kg to 1 Mg) blocks of cheese are made in a milk factory. When a consumer product is to be made from it, the cheddar is shredded, flavour and/or culture is added, and then using heat and pressure it is re-packed into consumer sized packages

In some cases Sodium Citrate is added to improve the hold-together of the cheese

Sodium Citrate is an old additive for "cooking cheese", and it can be made at home with baking soda and lemon juice. Gently heat the juice of half a lemon in a microwave safe bowl, then add 1/2 to 1 tsp of baking soda and heat until fully reacted (bubbles cease). Add about a cup of grated cheese (hard or soft) and repeat a gentle heat and stir cycle until it forms a smooth "sauce"

This will remain liquid while warm

If you let this cool you can mold it, or form "slices"

http://modernistcuisine.com/recipes/silky-smooth-macaroni-and-cheese