How is aspiration any different from the one & only wish or attachment that persons following the Buddhist path are allowed: the dream-like wish to attain dream-like enlightenment for the sake of all dream-like beings?
They are two different ways of approaching aspiration, or perhaps we could say two different emphases or ways of explaining nonduality, corresponding to the "two great chariots" of the Mahayana, Nagarjuna and Asanga. Put simply, emphasising the dream-like nature of all phenomena and beings is Nagarjuna's approach ("form is emptiness" / Madhyamaka / 2nd turning), and emphasising their infinite cosmic nature (as in the King of Aspiration Prayers) is Asanga's approach ("emptiness is form" / Uttaratantra-shastra and the Buddhanature teachings of the 3rd Turning).