Interpretation of a proportion as a treatment variable in a difference-in-differences specification

Say we observe outcomes for all U.S. states over many years. Some law/policy is introduced and affects some states but not others. The policy may start anytime throughout the year for some states; for example, it could be introduced in January or September of the year. The basic model looks something like this:

$$\mathrm{ln}(y_{st}) = \gamma_s + \lambda_t + \delta P_{st} + u_{st},$$

where a logged continuous outcome $$y$$ is regressed on state fixed effects, year fixed effects, and policy variable $$P_{st}$$. To be clear, $$P_{st}$$ is equal to $$0$$ in any state-year where the policy was not in effect, a value of 1 where the policy was in effect for the whole year, and a value equal to the portion of the year the policy was in effect otherwise. This turns the binary variable into a proportion. For example, in a year before a state's first "full year" under the new law $$P_{st}$$ could equal .25 or .75 depending upon the month the law was enacted. On the other hand, if the policy is removed later, I suppose the variable could equal a value between $$0$$ and $$1$$ in the year it concluded. This approach would involve the researcher inputting fractional values at the extremes (i.e., a state must have a full year of policy exposure to receive a value of 1).

I can't quite wrap my head around how to interpret such a variable. In essence, we turned the binary variable into some sort of "continuous" treatment, where you have a fraction of exposure before your first full year of treatment, but full exposure in the years the policy is in effect. Is my understanding correct?

Once we enter this into a linear model as a continuous predictor, how should we interpret this variable? I suppose I'm looking for insight into how I can compare this interpretation to the binary case.

See this NBER working paper (footnote 33) where they detail the coding of this variable.