You are quite OK in saying:
He's a person just like you and me.

agree with RES

Thank you for your answers of reassurance, the reason why I asked is because already twice I was corrected by a native English speaker that said that they say in the US "a person like me" and to say "like me and you" would sound awkward. How can this be????

For it to sound correct in ordinary American usage, it would be better put: 'a person like you and I' or 'somebody like us'.

If the native speaker actually told you 'a person like me' is correct, as opposed to 'a person like you and me', then that person was wrong. In a conversation where you directly address another speaker, and your rhetorical strategy is to point out some common ground between you and also say a third person shares that common ground, then you naturally include both yourself and the other person.

Interessante Behauptung, aber sonst auch nix.

Whatever that last German comment was intended to achieve, I am not sure. But there are several acceptable idiomatic renderings of 'ein Mensch wie Du und ich'.
The best would depend on context. Sometimes, in fact, you can replace the 'und' with an 'or' in American English: 'a person like you or I'.

Based on the original post, I'd say: 'a person just like you and me'. It is a well-known error in English to put yourself first. 'She and I are going to the movies' is correct while 'I and She are going to the movies' sounds clearly wrong.