The prefixed كَ in كَالصَّائِمِ expresses comparison: “like.” It is attached to the next word, الصَّائِمِ. الصَّابِرِ describes that fasting person as patient.
During fasting, patiently bearing hunger, thirst, and the discipline of restraint are ways in which patience appears. The grateful eater is compared with the patient faster to connect two virtues with two different situations.
Being like another person does not mean eating can replace fasting or that the reward of every meal and every fast is numerically identical. It is a moral comparison, not a substitution for a religious act.