Is there any Talmudic sources that claim the origin of the Hebrew word “dati” (דָּתִי) comes from the Avestan word daena?
דָּתִי is a recent coinage; its first attestation listed by the Hebrew Language Academy is from 1851, so there is no possibility of a Talmudic source on it.
The word is an adjective derived from the noun דָּת, which is much older, appearing already in the Bible in Esther (its Aramaic equivalent is also used in Ezra and Daniel). In the earlier books, it's entirely absent (if we follow the ketiv in Deuteronomy 33:2). The generally-accepted etymology is a loan word from Persian (dāt, dād, "law, justice").