No, no, no. Perfectly natural. He'll figure it out in time. No adults go around saying, "me am hungry." Plus, the examples you gave are pronouns, and he's using the personal pronoun (me/I) even though it's not exactly the correct one, and then possessive rather than the personal, but you're referring to a developmental issue. All children do this. Their brains are trying to figure out the whole grammar thing, so sometimes that universalize a rule, say, adding s's to all ending to make them plural (like "mices"), which means they've learned the rule, just not the exceptions or irregular forms. Here's a quote from an article online about children's language development:
"Making Corrections
Attempts to correct the form of a child's expression prove notoriously useless:
Child: want other one spoon, daddy
Father: you mean, you want the other spoon
Child: yes, I want the other one spoon, please daddy
Father: can you say 'the other spoon'?
Child: other ... one ... spoon
Father: say 'other'
Child: other
Father: 'spoon'
Child: spoon
Father: 'other spoon'
Child: other ... spoon. Now give me other one spoon?" (adapted from Braine 1971: 161)
And further down,
"Children seem to have taken the duchess' advice to heart: most recent research at every level of language study suggests that children are less concerned with "correctness" or with the form of their utterances, and much more concerned with communicating, with being understood, with making meaning" (de Villiers and de Villiers 1972; Halliday 1975 and 1978).
For the whole article, see http://papyr.com/hypertextbooks/grammar/lgdev.htm
Perfectly natural, and usually funny as well!