Borax (sodium tetraborate) it acts as the emulsifier. Borax is a naturally found mineral.
It's also a staple ingredient to almost all natural laundry detergent recipes.
It's borax + beeswax that is the emulsifier. Not borax alone. And not beeswax alone. The borax-beeswax emulsifier is unstable and prone to failure, which is why few people use it since there are more reliable emulsifiers available to home crafters.
Borax works as a water softener when used as an additive when for washing clothes, but washing soda is more effective than borax. Washing soda breaks down over time when included in a water-based mixture, however, which is why borax should be used, rather than washing soda, if a person has her heart set on making a liquid laundry soap mixture.
I know, I know -- washing soda, borax, and often baking soda are all included in a ton of the laundry soap mixes out on the internet, both wet and dry, but washing soda is the single best choice of the three, especially for a dry mixture. You can also use washing soda with a water-based soap mixture -- just add the dry washing soda separately. It's the combination of time + water that is the downfall of washing soda.
Save the baking soda for baking and the cat's litter box -- it's really not useful in a laundry soap mix.