Old short story that skewers political correctness
I recognized it as soon as I saw the title of your question. She skewered political correctness before it was invented, or at least before it ran amok.
OK, seem to recall reading it in an ANALOG collection if that helps.
You must have read in in a GALAXY collection: The Second Galaxy Reader of Science Fiction (edited by H. L. Gold), or maybe the abridged British reprint. The author is E. E. Smith—the other one, Evelyn E. Smith—and the title is "Tea Tray in the Sky". It originally appeared in Galaxy Science Fiction, September 1952; you can read it at the Internet Archive.
A young monk leaves the abbey he has lived at most of his life, and travels to New York City.
The Father Superior had smiled. "You are not yet a fully fledged Brother, Michael. You cannot enter your novitiate, until you've achieved your majority, and you won't be thirty for another five years. Why don't you spend some time outside and see how you like it?"
All the laws and taboos of the alien cultures are adopted by this civilization whenever they join the group, even if they make no biological sense or damage parts of other cultures.
"Fasten your suction disks, please," the stewardess, a pretty two-headed Denebian, ordered as she walked up and down the gangway. "We're coming into Portyork. I have an announcement to make to all passengers on behalf of the United Universe. Zosma was admitted into the Union early this morning."
All the passengers cheered.
"Since it is considered immodest on Zosma," she continued, "ever to appear with the heads bare, henceforward it will be tabu to be seen in public without some sort of head-covering.
Wild scrabbling sounds indicated that all the passengers were searching their packs for headgear. Michael unearthed a violet cap.
The salesman unfolded what looked like a medieval opera hat in piercingly bright green.
"Always got to keep on your toes," he whispered to the younger man. "The Universe is expanding every minute."
Examples include a species that considers showing hands in public to be offensive, so everyone wears gloves
Carpenter blushed and looked away. "Didn't you know that on Electra it is forbidden for anyone to appear in public with his hands bare?"
"Of course I know that," Michael said impatiently. "But what's that got to do with me?"
The salesman was wide-eyed. "But if it is forbidden on Electra, it becomes automatically prohibited here."
"But Electrans have eight fingers on each hand," Michael protested, "with two fingernails on each—all covered with green scales."
Carpenter drew himself up as far as it was possible to do so while lying down. "Do eight fingers make one a lesser Universal?"
"Of course not, but—"
"Is he inferior to you then because he has sixteen fingernails?"
"Certainly not, but—"
"Would you like to be called guilty of—" Carpenter paused before the dreaded word—"intolerance?"
"No, no, no!" Michael almost shrieked. It would be horrible for him to be arrested before he even had time to view Portyork. "I have lots of gloves in my pack," he babbled. "Lots and lots. I'll put some on right away."
(not yellow ones, that's a sacred color of another species),
Carpenter pressed his hands to his eyes. "Yellow is the color of death on Saturn, and you know how morbid the Saturnians are about passing away! No one ever wears yellow!"
the word "history' is considered obscene as several cultures have risen so rapidly they have none,
"I should have told you," Carpenter reproached himself as the Meropian swirled off. "Never mention the word 'history' in front of a Meropian. They rose from barbarism in one generation, and so they haven't any history at all. Naturally, they're sensitive about it."
and Times Square has been rearranged into a square and is full of clocks as another race finds anything not literal to be offensive.
"This," said Carpenter, "is Times Square. Once it wasn't really square, but it is contrary to Nekkarian custom to do, say, imply, or permit the existence of anything that isn't true, so when Nekkar entered the Union, we had to square off the place. And, of course, install the clocks. Finest clock museum in the Union, I understand."
The city is also saturated with constant advertising in the form of jingles from speakers - such gems as "Our pencils are perfect from tip to rubber, for the lead is from Yed and the wood from Dschubba".
A large scarlet pencil jumped merrily across the advideo screen. The face on the eraser opened its mouth and sang: "Our pencils are finest from point up to rubber, for the lead is from Yed, while the wood comes from Dschubba."
The Empire State Building has been converted to a public lavatory, the only one in the city:
Michael gazed at the Empire State Building with interest. It was in a remarkable state of preservation and looked just like the pictures in his history—in his books, except that none of them showed the huge golden sign "Public Washport" riding on its spire.
Attendants directed traffic from a large circular desk in the lobby. "Mercurians, seventy-eighth floor. A group Vegans, fourteenth floor right. B group, fourteenth floor left. C group, fifteenth floor right. D group, fifteenth floor left. Sirians, forty-ninth floor. Female humans fiftieth floor right, males, fiftieth floor left. Uranians, basement . . ."
Hordes of homesick Sirians sing their sad songs:
"Our wings were unfurled in a far distant world, out bodies are pain-racked, delirious. And never, it seems, will we see, save in dreams, the bright purple swamps of our Sirius."
"Foreign planets are strange and we're subject to mange. Foreign atmospheres prove deleterious. Only with our mind's eye can we sail through the sky to the bright purple swamps of our Sirius."
"When our minds have grown tired, when our lives have expired, when our sorrows no longer can weary us, let our ashes return, neatly packed in an urn, to the bright purple swamps of our Sirius."