Wow great find! I had no idea it was so similar to a real jingle.
For non-Japanese speakers: although the lyrics sound identical apart from the name, they are slightly different:
Family Mart version: "Anata to conbi ni, Family Mart" (In tandem with you, Family Mart)
vs
Tomato Convenience Store: "Anata no conbini, Tomato Mart" (Your convenience store, Tomato Mart).
Thank you :)
Yeah, it's pretty clever changing the words to make it sound so similar yet also making it have new meaning. But do we have official lyrics to the Tomato Mart jingle? Listening to the vocals in isolation, (will upload in time) it does kinda sound like it's saying あなたとコンビに、トマトマート rather than あなたのコンビニ、トマトマート unless I'm hearing things.
Also, another cool thing; The actual tomato logo is similar to one of Family Mart's original logos in use from 1981 to 1992.
It's a smiling red circle with a smiling star next to it. It kinda looks like a very stylized tomato, or even an anthropomorphized tomato.
Furthermore under the logo on the lower Ad, it says 「(アレフ)という名で、セゾングループは 国際花と緑の博覧会に出展しています。」
"Under the name (Aleph), Saison Group takes part in the International Garden and Greenery Exposition." which, according to Wikipedia "was the first large-scale international gardening exposition in Asia and focused on the theme of the "Harmonious Coexistence of Nature and Mankind."
https://en.m.wikipedia.org/wiki/Expo_'90
Further suggesting the logo may be a sort of stylized/anthropomorphized tomato. (tho as stated the logo was created in 1980s, the expo started in 1990s)
Edit: BONUS
Alternative but non watermarked version of the Family Mart logo history video.