The real connection between Thanksgiving and Chanukah is the Jewish Temple

From Congress' discussion of declaring a day of thanksgiving in 1789:

Elias Boudinot was a delegate from New Jersey; Roger Sherman was a delegate from Connecticut.

Roger Sherman naturally associated giving of thanks to the Biblical accounts of the celebrations of thanksgiving at the original dedication of the Temple.

Chanukah means "dedication" - celebrating the re-dedication of the Temple.

(From NYT quoting Rabbi Meir Soloveitchik, new rabbi of New York synagogue Shearith Israel, which was founded in 1654.)

(corrected date h/t Barbara)