1.3 Vs0 certainly applies to light aircraft as well as large airliners, although airliners use more complicated concepts as well, like 1.23Vsr0 for example.
As an aside, many light aircraft also need to fly at around 1.2Vs1 after take-off from a short runway. It's not specified in rules, but it does work. There are all sorts of relationships between speeds in most aircraft that arise simply because of common design characteristics across the worldwide fleet - engine power-to-weight ratio, wing aspect ratio, stability requirements and so on.