Is there a scientific or food industry term for what makes a water solution behave like a detergent, and perhaps an associated unit of measure?
For example, the pH of a solution measures its acidity; is there a similar term for "soapiness"?
Since I could see this being a cooking/food science question if you really wanted to play with this property of liquids in drinks, perhaps a quick answer is in order. This really would be quite well-suited to the physics stackexchange, though - this is a concept that's quite describable by physics.
To follow up on Sobachatina's comment, the property you're in fact looking for is surface tension (which is reduced by surfactants like detergent). If you want to compare liquids, be they water, water+detergent, milk (water+stuff), or even alcohol or oil, you just want to compare their surface tensions.
Surface tension is in fact a measurable quantity, with dimensions of force per length. There are a couple of equivalent definitions:
You should note that surface tension depends on the fluids on both sides of the surface. Most commonly we talk about surface tension of an arbitrary liquid against air, but surface tension against glass, plastic, or metal is commonly visible in your vessels and utensils, and surface tension between various fluids is of course visible when you try something like mixing oil and water.
As for actual examples, Wikipedia has a short data table with surface tensions of various fluids against air. Among them are a few potentially relevant for cooking: acetic acid + water (though stronger than normal vinegar), ethanol + water, a very concentrated sucrose solution, and water. Note that the addition of alcohol or acetic acid substantially reduces surface tension of water!
If anyone can follow up with more data on surface tension of liquids commonly seen in cooking, I'd be very interested!
It's also possible to measure surface tension at home, though I've never done it, and I'm not sure how accurate you could get.