Hi - I am happy to help. The fish flanking the submarine are called "dolphins" but not the mammals we think of. The original intent of the designer was to portray Mahi-Mahi, also know as "Dolphin Fish".
As to the origin/history of the insignia: The original suggestion for a submarine badge or distinguishing device came from Capt. Ernest King in early 1924 (he later became Commander in Chief, US Fleet during WWII - similar to the current Chief of Naval Operations). The Navy started working on designs of various types - even Capt. King submitted one which was rejected. The Navy asked a Philadelphia firm involved in designing Naval Academy class rings to give a try. That firm came up with 2 designs which were ultimately combined into the one we have today.
The final design went through the Bu. of Navigation (know as Bu. Personnel today) and on to the acting Sec. of the Navy, Teddy Roosevelt, Jr., who officially accepted the design as the official submarine qualification badge in March of 1924.
From 1924 - 1941 the badge was only authorized to be worn by officers and men while active in submarine units. Once transferred out of submarine service the badge could not be worn. The uniform regulations of 1941 changed that to permit the badge to be worn permanently once it was awarded.
Sorry there is no real significance (that I can find) to the dolphins other than the fact that it was a design submitted by a firm and accepted.
I see that I have rambled on a bit but I trust this helps. Thanks for your interest in the Naval History of our country.