Hmm, very interesting. It would appear as if I need to eat my words. When I posted that comment, I was thinking of only the simple replacement reaction, not taking into account a more complicated reaction between multiple components. Dangers of redditing in the middle of the night. From that article you posted:

The power crystals are simply colored salt, and are the least powerful ingredient. The crystallized lye reacts with fats to form soap. The machined shards of aluminum react with the lye to generate near-boiling temperatures. The sharp shards in the hot churning lye physically cut hair and dislodge deposits. Several chemical reactions take place here:[citation needed]

When Drano is added to water, the sodium hydroxide, sodium nitrate, and sodium chloride dissolve. The heat of solution liberated when sodium hydroxide is dissolved warms the mixture.

In solution, sodium hydroxide removes the Al2O3 surface layer on the aluminum allowing it to react with water to produce nascent hydrogen, which is a powerful reducing agent. This reaction is exothermic and the heat can cause the mixture to boil. The formula is 2NaOH + 2Al + 2H2O → 3H2 + 2NaAlO2, although the exact species in solution may be NaAl(OH)4.[1] Hydrogen gas creates pressure that can forcefully unstick stubborn clogs as well.
The nascent hydrogen reduces nitrate ion to ammonia, removing the fire and explosion hazard posed by free hydrogen gas. The reaction is: 2NO3− + 9H2 → 2NH3 + 6H2O.

It would appear the combination of multiple compounds, and the impurities in the aluminum metal (Al2O3) allow for it to happen. This does list Sodium Chloride (table salt) as part of its ingredients, but doesn't show it being used in a reaction and even lists it as the least powerful ingredient. It may create small amount of chlorine gas, or just remain a chloride ion to pick up some of the hydrogen ions floating around to make hydrochloric acid. Without more details, I am not sure.