In terms of the sound alone, both have "TH". But the endings, "AT" and "EM" feel quite different to me. "AT" reminds me of the "HATA" sound at the roots of hatha yoga.
In terms of congnative meaning, as I see it: "that" is one, "them" is many. However, "I am that" is like "I am one of them," as you have suggested, both have to do with a shift of perspective. Both suggest a witness.