I propose Jerome K. Jerome's "The New Utopia" (1891), which can be read online.
Story similarities
In this story, the narrator is a man who falls asleep and misses "the great social revolution of 1899", remaining asleep (and preserved in a glass case at the Museum of Curiosities) for one thousand years until he awakes in the 29th century. This far-distant future is a socialist one with all people being equal and all things decided by "THE MAJORITY", which is spoken of as a sort of god.
Everyone looks very similar, dressed in grey uniform and with short black hair:
All the people that we met wore a quiet grave expression, and were so much like each other as to give one the idea that they were all members of the same family. Everyone was dressed, as was also my guide, in a pair of grey trousers, and a grey tunic, buttoning tight round the neck and fastened round the waist by a belt. Each man was clean shaven, and each man had black hair.
Even men and women look the same, and instead of names, people have only numbers to identify them, which are displayed on a metal plate attached to their clothes:
“How do you know they are women?” I asked.
“Why, you see the metal numbers that everybody wears on their collar? [...] Well, the even numbers are women; the odd numbers are men.”
“How very simple,” I remarked. “I suppose after a little practice you can tell one sex from the other almost at a glance?”
“Oh yes,” he replied, “if you want to.”
[...]
“Don’t people have names, then?”
“No.”
“Why?”
“Oh! there was so much inequality in names. Some people were called Montmorency, and they looked down on the Smiths; and the Smythes did not like mixing with the Joneses: so, to save further bother, it was decided to abolish names altogether, and to give everybody a number.”
“Did the Montmorencys and the Smythes object.”
“Yes: but the Smiths and Joneses were in THE MAJORITY.”
“And did no the Ones and Twos look down upon the Threes and Fours, and so on?”
“At first, yes. But, with the abolition of wealth, numbers lost their value, except for industrial purposes and for double acrostics, and now No. 100 does not consider himself in any way superior to No. 1,000,000.”
People are washed by the state instead of washing themselves (so they would all smell similarly):
He said that they had found they could not maintain their equality when people were allowed to wash themselves. Some people washed three or four times a day, while others never touched soap and water from one year’s end to the other, and in consquence there got to be two distinct classes, the Clean and the Dirty. All the old class prejudices began to be revived. The clean despised the dirty, and the dirty hated the clean. So, to end dissension, the State decided to do the washing itself, and each citizen was now washed twice a day by government-appointed officials; and private washing was prohibited.
The story similarities are clear: physical uniformity, grey clothing, wearing metal numbers (whether it be on their sleeves or collars), washing, and hair are mentioned both in the future seen by Enoch Soames and that described in "The New Utopia". Furthermore, in both cases the character who finds himself in the future goes to a museum, and it is either one hundred or one thousand years ahead.
Historical context
Your description reminded me somewhat of the much more famous Nineteen Eighty-Four by George Orwell, which I knew was influenced by the moderately well-known We by Yevgeny Zamyatin. Perusing the Wikipedia page for the latter, I learned that We was in turn influenced by "The New Utopia". Nineteen Eighty-Four and We are of course too late to have influenced Beerbohm's 1916 story, but "The New Utopia" seems just right.
Socialist utopias in fiction have a long history, with some prominent examples being listed here. Several notable such books were written around the end of the 19th century and the start of the 20th, including Edward Bellamy's Looking Backward (1888) and H. G. Wells's A Modern Utopia (1905). But neither of these has anywhere near as many common elements with Beerbohm's vision as those listed above for "The New Utopia". The latter is by far the most convincing match I've been able to find.
Also, we know for a fact that Beerbohm was aware of Jerome and his work - indeed, he was known as a critic of Jerome. Wikipedia mentions that Beerbohm called Jerome a "tenth-rate writer", and an Amazon review mentions that "Max Beerbohm has the reputation of being a pacific personality (with the exception of the poor Jerome K. Jerome)". So it's plausible that he would have written "Enoch Soames" with Jerome's utopia specifically in mind.
There is a possibility that both Beerbohm and Jerome were basing their "utopias" on some previous work. A bachelor's thesis about Jerome's writings and social/political views (PDF link) describes "The New Utopia" as connected purely to Jerome's own political views, without reference to any previous work, but Brian Stableford's Scientific Romance: An International Anthology of Pioneering Science Fiction says that:
Jerome's "The New Utopia" (1891) was, like ["June 1993" by Julian Hawthorne], a response to Edward Bellamy, but is more farcically parodic, in keeping with his usual manner.
And about the other story:
"June 1993", which appeared in the February 1893 issue of Cosmopolitan, is part of a flood of utopia speculations that followed the unexpected but enormous success of Edward Bellamy's Looking Backward, 2000-1887 (1888). Most of the responses focused on the novel's prospectus for socialist economic reform - far more controversial in rampantly capitalist America than in Europe, where France had a thriving subgenre of anarchist utopian fiction - but Hawthorne focuses on the side-effects of technological advancement.
There was indeed a massive response to Bellamy's Looking Backward in other stories, both positive and negative takes on his vision of the future. Bellamy's novel also concerns a man who falls asleep at the end of the nineteenth century and wakes up at the end of the twentieth, and it seems clear enough that Jerome's story is a direct (satirical) response to Bellamy's. However, I still contend that the similarities between Beerbohm's future and Jerome's are far greater than with Bellamy's, and that Beerbohm was specifically making use of Jerome's reimagining of Bellamy's socialist utopia, rather than being directly inspired by Bellamy himself.
In particular, I quote some passages of Bellamy's story which are startlingly different from Jerome's and Beerbohm's more negative visions of socialism:
It did not appear that any very startling revolution in men's attire had been among the great changes my host had spoken of, for, barring a few details, my new habiliments did not puzzle me at all.
[...]
"Because," Edith explained, "although the income is the same, personal taste determines how the individual shall spend it. Some like fine horses; others, like myself, prefer pretty clothes; and still others want an elaborate table. The rents which the nation receives for these houses vary, according to size, elegance, and location, so that everybody can find something to suit. The larger houses are usually occupied by large families, in which there are several to contribute to the rent; while small families, like ours, find smaller houses more convenient and economical. It is a matter of taste and convenience wholly. I have read that in old times people often kept up establishments and did other things which they could not afford for ostentation, to make people think them richer than they were. Was it really so, Mr. West?"
In Bellamy's future, people have individual names (not numbers) and freedom of individual expression in things like clothing and hairstyle. In Jerome's future, society is much more rigidly restricted. Recall that Jerome's story is believed to have inspired We and indirectly Nineteen Eighty-Four; it's not implausible that it was being directly referenced closer to its own time too. Indeed, the publication date of "Enoch Soames" is only a few years before We.
