This may be a naive question, but here goes. If I have a set of empirical data and fit a kernel density to it, and then obtain a new single value which possibly comes from the same process which generated the original data set, can I assign a probability that this new value belongs to the set/process by simply reading the value off the y axis where the new value on the x axis intersects the kernel density line and dividing by the area under the density line?
No, I'm afraid not. The kernel density estimand is the probability density function. The y-value is an estimate of the probability density at that value of x, so the area under the curve between x1 and x2 estimates the probability of the random variable X falling between x1 and x2, assuming that X was generated by the same process that generated the data which you fed into the kernel density estimate. The kernel density estimate doesn't say anything about the probability a new value was generated by the same process.