Why isn't there a standard unit of promoter strength?
Transcription rates are very context dependent. This shouldn't surprise you, since the same genes aren't expressed in your lungs and your eyes. They're also very environment dependent, this shouldn't surprise you either, since it's pretty obvious every organism is going to respond to temperature, their current nutritional state and so on.

So, in order to define promoter strength, you'd first need to define the exact conditions under which you're measuring it, including the exact strain you're measuring it in. Because if you get any of those even slightly different you're going to get a different answer. Doing this might get you a consistent number but you're abstracting away the most important parts of differences in transcription anyway: how it varies between cells and contexts.

Moreover, even if got past these hurdles you'd not have a particularly useful answer anyway since the rate of transcription is only a small part of the whole process. Why not consider mRNA durability, or rates of translation, or protein durability? What matters very much depends on what you're considering.