Yes they did, and so did many nation's generals. Hence why they are collectively known in some places as "Flag Officers". Specifically for admirals at sea their ship will fly a special flag from the mast that corresponds tot heir rank, hence the term flagship! In the USN the flags are pretty simple, corresponding to the number of stars an admiral has.

In battle it was usually at the admiral's discretion when underway to fly it or not, so as to avoid making the ship conspicuous, and other substitutes could also be used. Though in days of old and indeed even often into the 20th century it was simple expected, and in the age of sail it was often more important to be able to spot the flagship and know if she was still fighting then try to deny that information to the foe in the confusion of battle. Though would ping /u/jschooltiger and /u/thefourthmaninaboat for their knowledge of the evolution of the practice in the Royal Navy that influenced so much of the rest of the world.

But in peacetime they are perfectly common, though he modern has many fewer seagoing flag officers than it once did so it is less common. But as soon as the admiral comes aboard his flag is supposed to be broken out on the rear most mast, and when a senior admiral, officer, or SECNAV or Assistant secretary coem aboard their flag takes precedence. Which made things very hectic on the Missouri in Tokyo Bay when a 4 star admiral, 5 star admiral, and 5 star general as senior representative all came aboard. It also leads to situations of juggling halyards, here is the cruiser Houston with both FDR and a 4 star aboard flying both flags at San Diego.

For a period appropriate idea of what it would look like, for the USN at least, we can look to the dear old USS Pennsylvania, which spent 20 years as the flagship of the Battle Force, Atlantic Fleet, and Pacific Fleet before Pearl Harbor. Here she is in 1921 for the Fleet Problems off Cuba flying the 4 star flag of ADM Henry Wilson. Here she is during the same summer flying the 4 star and anchor flag of the Secretary of the Navy Josephus Daniels off Panama.

It was not a strict requirement in the face of operation security though as mentioned. Here is the Augusta taking Truman across the Atlantic to Potsdam and you note she isnt flying the President's ensign.