I recently learned in a class that we consider Eisav Jewish. This is corroborated by most of the answers here. I have also learned that we customarily wish someone "Yiddishe nachas" (rather than just "nachas") to distinguish these from "goyishe nachas"; that is, the nachas of Eisav (hunting, etc.) But why should we see Eisav's nachas as goyish if he were in fact Jewish?
The question is predicated on a joke which you missed or which wasn't properly explained.
One of the grandsons of Esav was named Nachas, as we see in Bereishit 36:13:
וְאֵלֶּה בְּנֵי רְעוּאֵל, נַחַת וָזֶרַח שַׁמָּה וּמִזָּה; אֵלֶּה הָיוּ, בְּנֵי בָשְׂמַת אֵשֶׁת עֵשָׂו.
And these are the sons of Reuel: Nahath, and Zerah, Shammah, and Mizzah. These were the sons of Basemath Esau's wife.
The joke goes that when specifying Nachas, we mean the Jewish form of Nachas, and not the Nachas who was the grandson of Esav.
A joke like this does not need justification and need not subject itself to analysis, but if you still desire it, we can point out that even if Esav would be considered Jewish, his grandson was not.
The development of the term seems not related to Eisav particularly,
Another common use of the word naches is in characterizing something as particularly Jewish or non-Jewish. If one wishes to say that a mode of entertainment or pleasure inducement is particularly goyish (i.e., no Jew would find it entertaining or pleasurable), one would call it goyishe naches (גוישע נחת). (See Max Weinreich’s treatment of the phrase “goyim-naches,” abbreviated to GN, among German Jews based on its medieval German pietistic origins in the Sefer Chasidim. See also Dovid Katz, who writes that the traditional Yiddish term for hunting was goyishker naches (גוישקער נחת), since it was construed as a particularly goyish pastime.)