Of course. Many of us have used washing powder when experimenting, I had to use it on one occasion when miles from home and without a chemical in the van, having driven away with them sitting in the drive. Excellent result, but I spent a lot of time rinsing, just in case !
As long as the water is hot and you are able to get good dilution, it will work................but domestic cleaners are bulked up and I'm sure would wear away jets pretty quickly, also, as already mentioned they need more rinsing .
In an emergency, I'd use what's available.