means Soft and pulpy to the point of losing its shape, or excessively sentimental and tender in an emotional way.

from Born from 'mush,' the thick cornmeal porridge whose name is likely a softened variant of 'mash' — that which has been crushed into a yielding pulp. The '-y' tacked on turns the stuff into a quality, and somewhere along the way the word slid from describing overcooked food to describing overcooked emotions, the kind that go soft and shapeless when love gets involved.