Aready an entry (1058900).

sankoku:
[1] [題名などで]主題を示す「ー商社」
[2]  その名にもっともふさわしいことをしめす。これこそ。「まさにー駅伝」

(daijs)
https://www.weblio.jp/content/the#:~:text=ザ【the】&text=《英語の定冠詞から》名詞,商社」などと用いられる。
ザ【the】
読み方：ざ
《英語の定冠詞から》名詞に付けて、その語のもつ性質・機能などを強調したり、普通名詞をその典型を表す固有名詞のように扱ったりする。広告文や映画・テレビ番組・雑誌記事のタイトルなどに「ザ‐バーゲン」「ザ‐商社」などと用いられる。

The Ramen屋
https://tabelog.com/kanagawa/A1401/A140104/14008330/
The chocolate 明治 ザ・チョコレート
https://www.meiji.co.jp/products/brand/the-chocolate/

Calbee:The potato (chips) ザ・ポテト オホーツクの塩味
https://www.calbee.co.jp/newsrelease/200402b.php

Competing boutique the-potato
https://the-chocola.com/lp?u=potato
Also sells "the chocola" "the pudding", the tiramisu:
https://the-chocola.com/shop/product_categories/sweets
The Icecream
https://qina.co.jp/products/icecream
The ICE (cream)
https://furusato.jal.co.jp/goods/detail/0318290cc75cdad84294b98961763d90
The caramel　ザキャラメル
https://www.haagen-dazs.co.jp/products/special/the-caramel/

(got here from 決定版)
Kind of surprised there is no entry for "the".  Not sure how to pair ザ, ざ, and "The", which are all perfect Japanese(in marketing).

I'd be tempted to note this as "marketing fluff".

It's pervasive in the grocery store, and as a westerner, somewhat hilarious.

Written in English more often than not, I think. (but prevalent in kana as well)

Open to suggestions, just throwing this out there at the moment.  It's in daijs and sankoku, so I think it should qualify, and its meaning is also completely non-obvious (until you've spent time in Japan).  "The Choya" is the one you can't miss on public transportation, and unlike the others above, it doesn't appear to highlight a category of product, but rather the finest of *their own* product.  Choya makes all kinds of Umeshu (plum wine), and "The Choya" is the line of their higher-end (well, $10~$15+/bottle) product.

https://www.choya.co.jp/products/

It's not that exclusive of a club - a large % of their products are "The Choya" now...
https://www.choya.co.jp/brand/the_choya_special/