Hi,
Since 'doing' is dealing with a human sense - i.e., one's mental state, or one's physical state - then you could use the adjective 'good' to describe the verb doing. So if you are feeling happy, you could say "I am doing good"; and if you are recovering from surgery, you could say "I am well."
When you say 'I am well', 'well' is an adjective describing the pronoun 'I'. By definition, an adjective cannot describe a verb.
When you say 'I am doing well', 'well ' is an adverb that modifies the verb 'do'.
Thus, 'I am doing good' is considered substandard because you are using an adjective to modify a verb. You will hear some people say it, but I wouldn't recommend it. For example, don't say it in your job interview for a position where a good education is an important requirement.
Bet wishes, Clive