Because doing is a verb and verbs, in general, are modified by adverbs (such as well) and not by adjectives (such as good).
The boundaries are a bit fuzzy sometimes, and well can be an adjective as well ("well-mother checkup", for example), and some verbs do take an adjective complement in some cases, usually with a different meaning to taking an adverb (e.g. "the dog smells good" = a good smell emanates from the dog vs. "the dog smells well" = the dog has a keen sense of smell and can perceive smells efficiently).
With "feel", for example, you also have "I feel well" = I have a good sense of touch vs. "I feel good" = I am in good health.
"do" is not (yet) in this group in Standard Written English, so "I'm going good" is not good SWE (yet), at least not in the sense "be in good health". This may change eventually as usage shifts, but not yet. (It is grammatical in the sense of "perform good deeds", as Alex points out below.)
In general, Duo doesn't accept language that is too colloquial, no matter how many people say things such as "You ain't seen nothing yet" or "I'm doing good". I'm not even sure about regionalisms such as "My car needs fixed" or "We go to the Grand Canyon anymore".
Native English speakers resort to slang more often. I slipped right into Australian English slang for this and typed "My cat is going good" without thinking. Is this used in Australia? Yeah. A major greeting for us is "Hi, how you going?" and a response could be "going good, thanks". But it's not great English. LOL.
If one is asked, ¨How are you?¨ the proper answer is good, and not well. This is due to it describing a state of being. However if the question is, ¨How are you doing?¨, the answer can be well since this is an active rather than a stative. verb or condition. Yes, it true, that usage is commonly ¨well¨. If the dictionary/grammar book, is to be helpful it should be both prescriptive (rules) and descriptive (usage), and that is why we approach the issue differently. Personally, the prescriptive approach is preferred, and that preference stems from my vocation, a university English Professor.