“Soup and fish” was a traditional first course in the formal dinners of the 19th century. “Soup and fish” (also “soup-and-fish”) also became a nickname for a man’s dinner suit. The Oxford English Dictionary records a similar term, “soup-and-pattie,” from 1829.
 
“Soup and fish” is cited in American slang from at least 1909. The term—like the “soup and fish” courses—is not often used today.
 
   
World Wide Words
SOUP AND FISH
[Q] From Lee-Ann Nelson: “I am baffled by an expression from P G Wodehouse. Bertie puts on his soup and fish. Can you explain this?”
[A] I can. The soup and fish is a man’s evening dress, dinner suit, or dress suit, though I should really instead refer to it as a tuxedo, since — despite Bertie Wooster’s mainly London milieu — the phrase seems to be natively American.
 
Until I went delving in old US newspapers, I thought that Wodehouse had invented it. Indeed, the OED gives him the credit for its first use, in Piccadilly Jim in 1918: “He took me to supper at some swell joint where they all had the soup-and-fish on but me. I felt like a dirty deuce in a clean deck.” But there are earlier examples, such as this from The Atlanta Constitution of November 1914, in a report about local kids being given a slap-up meal by the Rotary Club: “There’s going to be no ‘fess up’ business; no ‘soup and fish’ outfits. It’ll be just a good dinner.”
 
But why soup and fish? Well, one dons these duds for a special occasion such as a formal meal. This is likely to be a heavyweight event, with many courses, starting with soup and followed by fish before one gets to the main event of the meat course. So the soup-and-fish is what one wears to consume the soup and fish.
   
Dictionary.com
soup-and-fish 
–noun Informal. a man’s formal evening clothes.
Origin:
alluding to the early courses of a formal dinner
 
(Oxford English Dictionary)
soup
In combination with other ns., as soup-and-blanket, soup-and-bully, soup-and-patty; soup-and-fish slang, men’s evening dress, a dinner suit.
1829 SYD. SMITH Let. in Lady Holland Memoir (1855) II. 299 He had not his usual soup-and-pattie look.
1862 DICKENS Somebody’s Luggage 26 She’d have no more chance again the ice, than a chaney cup again a soup-and-bully tin.
1900 Westm. Gaz. 26 Sept. 8/1 Making ground with his electors through the medium of the ‘soup and blanket brigade’.
1918 WODEHOUSE Piccadilly Jim i. 26 He took me to supper at some swell joint where they all had the soup-and-fish on but me.
1945 ‘A. GILBERT’ Black Stage xi. 149 What do you do about dinner here? Soup-and-fish or just a clean collar?
1970 H. MCLEAVE Question of Negligence (1973) xviii. 141 Get him to take off his soup-and-fish and show us his scar.
 
13 September 1910, Daily Herald (MS), “The Silver Horde” by Rex Beach (copyright 1909, Harper & Brothers), pg. 2:
“She fell for the name all right, but there must have been something phony about the clothes. That’s the trouble with this park harness. If I’d wore my soup and fish and my two gallon evening hat I’d have passed for a gentleman sure.”
 
18 June 1911, Duluth (MN) News-Tribune, sporting section, pg. 1:
Included in the outfit is the customary Tuxedo, together with a full regalia of the “soup and fish” raiment.
       
12 July 1914, New Orleans (LA) Times Picayune,  “The New Fable” by George Ade, pg. 25:
In his Afternoon Wear he resembled the Manager of a Black-Goods Department.
 
After donning the complete Soup and Fish, known in swozzey circles as Thirteen and the Odd, he didn’t look as much like a Waiter as one might have supposed. He looked more like the ‘Bus who takes away the Dishes.
 
Google Books
The Commuters:
A Comedy in Four Acts
By James Forbes
New York, NY: Samuel French
1910, 1916 (two copyright dates)
Pg. 52:
SAMMY. You don’t mind if I shake the benny and the soup and fish. (HETTY laughs) I’m a fancy little person when I’m all dolled up.
   
Google Books
Film Folk:
“Close-ups” of the men, women, and children who make the “movies”
By Rob Wagner
New York, NY: The Century Co.
1918
Pg. 215:
A fella like “Dress-suit Charlie,” for instance, has almost a clairvoyant hunch where they are goin’ to do ball-room stuff; and, as he is a doll-baby and a good dancer, he’s probably atmosphered in more society pictures than any fella in the country. The “soup and fish,” as we call the society stuff, pays five dollars a day.
 
Google Books
September 1918, Century Magazine, pg. 645:
The town was full of stories of how Mrs. Furness cooked in an evening gown, and Albert Edward, her husband, after dinner lighted a postprandial clay pipe of Army Cut, and smoked solemnly in his “soup-and-fish.”
 
24 January 1919, Chicago (IL) Daily Tribune, pg. 1:
‘SOUP AND FISH’ DAZZLES POLICE
Exquisite in Dress Suit, Seized as “Phony” Check Passer