The four letters hide a very specific, very small beginning: a dozen shop workers in a London room.
On 6 June 1844, George Williams, a draper, and eleven friends founded the Young Men's Christian Association. Britannica says the club was formed for the improvement of the spiritual condition of young men in the drapery and other trades. Williams worked for a drapery house and later became its head. The name described exactly who the group was for: young men, Christian in outlook, joined as an association.
The sources checked do not say who picked the name. What they do show is how quickly the name traveled. Similar clubs spread across the United Kingdom, reached Australia in 1850, and reached North America in 1851. The first North American branch was in Montreal and the second in Boston.
Outside English, the name changed with the audience. Wikipedia notes that Germany's organization changed its name in 1985 to show it was open to women as well as men. In 2011 Sweden's association began using KFUM Sverige, where the M now stands for the Swedish word for people rather than men. The women's counterpart, the YWCA, is a separate organization, although some local and national groups have merged with it.
The famous nickname is more recent than most people think. On 12 July 2010, YMCA of the USA adopted "the Y" as its brand name and redesigned its logo. Some branches kept the full initials in their local names.
The initials also became a song. The Village People released "Y.M.C.A." in 1978, and Wikipedia cites it as an example of how YMCA rooms had become a well-known, cheap and safe place to stay in unfamiliar cities.
The group's inverted red triangle emblem, introduced by Luther Gulick in 1891, is a separate symbol. It is not meant to be the letter Y.