Dawn's main ingredients are sodium laureth sulfate and sodium lauryl sulfate.
Liquinox uses sodium dodecylbenzenesulfonate.
They aren't chemically identical, but their detergent properties in water (solubility and critical micelle concentration) are similar enough to make them easily interchangeable. For cleaning grunge (finger prints and dust) from vinyl records, both get the job done. If you have to perform sensitive chemical analysis on lab glassware, where contaminants in the range of 1 part/million are too much, then use Liquinox.
Don't use as much detergent as shown in that video. Maybe 1% of what that guy used is all that's needed. Excess detergent is only harder to rinse away.