It appears that the food was noted as "lamb stomach" (haggis or similar) at the Studio Ghibli Layout Designs exhibitions.
The "fins" are where the haggis is sewn or tied.
On page 220 of the exhibition catalogue, the following description is given about the food in question:
While the description doesn't tell us the name of the food, it likens it to, and probably is, a lamb stomach.
Then on September 19, 2020, Spirited Away's key animator, Hiromasa Yonebayashi, tweeted that the food as written in the storyboard was Coelacanth's stomach (thanks to кяαzєя for bringing this to my attention). However, in the responses to the tweet, people pointed out that this information was never published in Studio Ghibli Storyboards Volume 13: Spirited Away, and what was published, in the aforementioned Studio Ghibli Layout Designs, contradicts his answer. Considering that in the anime production process, the layout designs come after the storyboards, Yonebayashi's answer recited from memory might have been revised later on. In light of the observations made above, and of the available images of haggis and a hypothetical dish of Coelacanth's stomach, I am more inclined to believe that the food Chihiro's father was eating was lamb stomach.
By the way, you can see the reconstructed food on display at the Ghibli Park.
Perhaps you've been told that the food is ba-wan, a Taiwanese street food. Perhaps you've been told that the town from Spirited Away is modeled after Jiufen, a tourist attraction in Taiwan. They are at best misinformation and at worst disinformation propagated by travel agencies and shopkeepers to increase sales. Hayao Miyazaki has denied twice, once in the TVBS news interview, and once more in the CTi news interview that Jiufen was an inspiration for the film's settings. What were really the inspirations and/or models for the settings are listed in this well-researched article: the Dougo Onsen, the Edo-Tokyo Open Air Architectural Museum and Meguro Gajoen being the officially confirmed places.