Well is an adverb, so it works grammatically. Good is an adjective, so the second one doesn't really follow the rules of grammar if you're describing your current state. But you will hear natives say it that way often. You could say "I'm doing good" to mean that you're bringing about good things by performing positive actions and it would be okay grammatically.

The main difference between "I'm doing well" and "I'm doing good" is their grammatical function and the context in which they are used. "I'm doing well" is grammatically correct and commonly used to indicate good health, good progress, or a good situation. For instance, "I'm doing well in school" or "I'm doing well after my surgery." "I'm doing good" is grammatically incorrect but can be commonly used informally to indicate that you are doing good things or doing good deeds for others. For instance, "I'm doing good by volunteering at the shelter" or "I'm doing good by donating to charity." In short, "I'm doing well" is used to indicate a good state of being, whereas "I'm doing good" is used to indicate doing good things for others.