I haven't watched Arianne's video to see how she packages it. The Humblebee version is packaged in an open jar, and you swipe your damp cleaning sponge or cloth over the detergent as you need it as you clean. This will contaminate the surface of the detergent with food particles and general grime as well as bacteria and fungi.
We've had similar discussions about other paste-type products and whether a preservative is necessary. For sugar scrubs, I think the general consensus is yes, preservative is important in scrubs. These scrubs are usually packaged in open jars so the user can easily use damp fingers into the container to remove the product and the mix of sugar and fat with a bit of water makes a great buffet for microbes to feast on.
Another paste-type product is the firm KOH soap paste (the thick gel that you dilute to make pourable liquid soap) that some people use for household cleaning. The consensus there is that preservative is not strictly necessary for KOH soap paste, because true lye-based soap is alkaline by nature and that alkalinity discourages microbial growth.
The detergent cleaner that Arianne and Humblebee are sharing is going to be alkaline to some degree, although the alkalinity of the finished paste will vary depending on the types of detergents you choose. I doubt the detergent cleaner will be as alkaline as KOH soap paste, however, so I'm not as confident that a detergent paste can be self preserving the way a soap paste can be.
Because most of us don't have a microbiology testing lab to know for sure, we have to make educated guesses about things like this. IMO, for personal use, it's up to you to make the call whether to use preservative or not. Again IMO, a preservative is needed if you plan to sell or give this product to others. But in the end, the decision is yours to make.