One ordinarily varies the concentration of substrate, not enzyme, in order to determine the KM. The value of KM is expected to vary with pH and might vary with ionic strength. With alkaline phosphatase, the choice of buffer can also influence the values of the kinetic constants because of an additional complication: the buffer Tris, for example, is an alternate acceptor of the phosphoryl group (the normal acceptor is water). One would have to see the article in order to evaluate the solution conditions.
It is not always true that the KM is a measure of affinity between enzyme and substrate in the same way that the dissociation constant is. The KM may or may not be equal to Kd; it depends on the enzyme and the substrate in question.