D-Day Operation Overlord: History Behind the Design
About the flag on the front of the shirt
From a news story when the flag was accepted by the President on behalf of the American people in July 2019:
"On June 6, 1944, the flag flew aboard the landing craft Control 60. Commanding the ship was a young Navy lieutenant, Howard Vander Beek, who was two days shy of his 27th birthday, Trump said.
"'Amid treacherous German minefields, raging winds and rough seas, Lt. Vander Beek and his crew led an astonishing 19 waves of American troops and equipment to those very dangerous beaches,' the president recounted. 'Through it all, this flag soared proudly above the waters of the English Channel, announcing the arrival of our American warriors.'
"After completing his mission on D-Day, Vander Beek took the flag — by then bearing the scars of German machine-gun fire and stains from dirt, diesel fuel and blood — and carried it with him in his backpack for the remainder of the war. He kept the flag until his death in 2014, Trump said.
"'Soon after, the flag was purchased at auction by Mr. Kreuk and Mr. Schols, whose relatives were among the hundreds of Dutch who perished in the German bombings of Rotterdam in 1940,' the president said. 'These two gentlemen paid half a million dollars to obtain the flag, just so they could return it as a gift to the American people and to the United States of America.’"
Source: U.S. Department of Defense new story at Defense.gov.
How were the beaches named?
"On the American side, the names chosen correspond to a state, Utah, and to a city Nebraska, Omaha. They were selected at random : At the moment when the operations were being named, a general asked two NCOs where they were from. For British and Canadians, the names correspond to the abbreviations for certain types fish : goldfish, swordfish and jellyfish. Jelly was changed to Juno in honour of the wife of one of the officers.”
Comment: My guess is that the fellow from Utah was from Salt Lake City, making the single word name of the state a better choice.
About the photo of men walking through the waves to the beach on D-Day
"Into the Jaws of Death - U.S. Troops wading through water and Nazi gunfire" A LCVP (Landing Craft, Vehicle, Personnel) from the U.S. Coast Guard-manned USS Samuel Chase disembarks troops of Company E, 16th Infantry, 1st Infantry Division (the Big Red One) wading onto the Easy Red section of Omaha Beach (Calvados, Basse-Normandie, France) on the morning of June 6, 1944. American soldiers encountered the newly formed German 352nd Division when landing. During the initial landing two-thirds of Company E became casualties. Photographer: Chief Photographer's Mate (CPHoM) Robert F. Sargent.
Sources: The National Archives and Records Administration and Wikipedia.