Yes, both me and à moi can act as indirect objects, and me can act as both a direct and indirect object: Tu me donnes might mean 'You give me (to someone)' or 'You give to me (something)', e.g.: Tu me donnes aux flics or Tu me donnes la main, Tu me donnes à manger.
Tu donnes à moi is odd but is common with a direct object pronoun before the verb: Tu le donnes à moi: 'You give it to me', it emphasizes the indirect object, the recipient here. The form Tu me le donnes is more common and neutral.
You can also use both forms at the same time, e.g.: Tu me le donne à moi ? (for example to double check you correctly hear it) or Tu lui donne à elle ! (to clarify it is not à lui but à elle or to emphazise your disagreement.)