Some features of the Thanksgiving celebration are reminiscent of Jewish/Biblical holidays.
1 — Thanksgiving commemorates a historical event: The landing of the Pilgrims at Plymouth Rock and their survival during their first year in the “New World.”
Biblical holidays — and the later “rabbinic” ones — almost uniformly commemorate specific events. Torah is clear about why the annual celebrations are to take place. Where the “event” might be unclear — as with Sh’vu’ot, for example — the rabbis, by analyzing the Torah-text, found a reasonable connection in receiving Torah at Mt. Sinai. The text of “Esther” explicitly states that the Jews are to have an annual commemoration. Whatever other allegorical or kabbalistic understandings are derived from these annual celebrations, their “p’shat” — their plain meaning — is to remind us of something that took place.
It’s also true that the date of the Thanksgiving celebration wasn’t firmly set until Lincoln’s time as the final Thursday in November. So the holiday commemorates an event, without necessarily being a strict “anniversary.”
2 — The concept of a free-will “thank-offering” is itself quite biblical. Although it certainly exists in other traditions, too, the Pilgrims very self-consciously identified themselves with “biblical” religious practices, especially ones that predated the time of the Mishkan and Temples, when the forms of worship were more fixed. It was therefore “natural” for them to show gratitude by a ritual sacrifice — in this case, of a turkey — for G-d.
One must point out that if G-d providentially preserved the Pilgrims through their life-threatening Atlantic voyage and the first winter they endured without adequate shelter or food, we should feel no less compelled to thank the Native Americans in general as G-d’s “messengers” — G-d’s “angels,” as it were — without whom the Pilgrims would not have survived.
Such free-will offerings are mentioned in the narratives of B’reishith/Genesis. These offerings could be done anywhere, at any time, even spontaneously, as one felt the need to do so. Therefore, again, the “Pilgim-ic” sacrifice done out in the open, not in a church, and not as part of a formal liturgical service, is reminiscent of those “biblical” sacrifices performed by Noah, Abraham, etc. Think of the ram that Abraham sacrificed in place of Isaac.
3 — The offering of an animal — a turkey — is again very reminiscent of biblical worship, which included animal-sacrifice. This was a one-time practice for the Pilgrims, not found at all in other Christian church traditions (but found once yearly in Islam). While the early Christians in Jerusalem might have continued to participate in animal-sacrifice at the Temple, other Christian groups, who later became the majority, ceased to do so — especially after the cessation of Temple worship altogether. While animal-sacrifice still had a symbolic meaning in Christian commentary and interpretation, the act itself early ceased to be part of Christian observance. 
4 — The “thank-offering,” or “thanksgiving offering” — the Korban Todah — was sacrificed in the Temple, but eaten anywhere in Jerusalem, presumably as a shared meal:
“The thanksgiving offering and ram of the Nazirite are kodashim kalim [of lesser strictness]. They are slaughtered anywhere in the Temple courtyard…[and] are eaten throughout the city [of Jerusalem], by anyone, during that day and night until midnight. 
Factually, turkey wasn’t actual eaten at the first Thanksgiving:
But the emphasis on a “shared meal,” so characteristic of the American holiday, is, again, reminiscent of a biblical model — in this case, of the “thank-offering.”
All Jewish holidays have an element of “thanksgiving” about them, but I’d offer that “Pesach/Passover” most closely resembles the American holiday in intent.
 a good article on the echo of the “Todah” — the “thank-offering” — in Catholicism and Christianity in general:
 Talmud: Mishnah Zevachim 5:6