When did music groups first start adopting names, such as The Beatles? Did Medieval musicians band together and adopt such names or is this some more modern development?

'First' questions are sometimes hard for one flair to answer, because they require you to have quite a wide range of knowledge - you'll see by the wording on my flair that I know the 20th century, and will leave anything before that to someone else. Additionally, the answer depends to some extent on what exactly you mean by 'music groups' and 'names such as the Beatles'.

So, assuming that by 'names such as the Beatles', you're referring to musical groups that have abstract names which are not immediately relevant to the group's music (or their location), which aren't simply backing bands for a named singer (e.g., in opposition to 'the London Symphony Orchestra' or to 'Joe Bloggs and his Swing Orchestra'), it certainly predates rock'n'roll acts like the Beatles, and almost certainly the 20th century.

In A New History Of Jazz Alyn Shipton refers to turn-of-the-century string bands in New Orleans as precursors to jazz (who had jazz attitudes and playing styles, but were more string-instrument based rather than brass based), and names groups like the Big Four, the Excelsior, the Tio and Dublais Orchestra and the Union String Band. The Big Four and the Excelsior might count in terms of bands as abstract as the Beatles, but might also be basically descriptive (e.g., the Excelsior might be the name of the venue where they got their reputation). Shipton alsao discusses a ragtime band who were friends with Scott Joplin called, at first, the 'Musical Bumpers', and then 'the Five Spillers' in 1907 (Spiller being the surname of the band leader and his wife the singer, so perhaps that doesn't count) who, by the time the jazz fad started in the mid-1910s were labelling themselves as a 'jass' band. Broadly speaking, however, 1910s or 1920s jazz bands typically highlighted a band leader - Jelly Roll Morton's Red Hot Peppers, for example, or the names were self-explanatory in terms of their sound (e.g., the Original Dixieland Jazz Orchestra) or where they were formed (e.g., the Casa Loma Orchestra). The original recording of 'Mood Indigo' by Duke Ellington and his band - usually called 'Duke Ellington & His Orchestra' or 'Duke Ellington & His Washingtonians' on most of their records - was released under the name 'The Jungle Band', which is meant to be evocative, but is at least somewhat abstract. Outside of jazz, the 1900s sees the first recordings of a group calling themselves the Peerless Quartet(te), which is getting on the way to abstract (though clearly the band title concretely refers to the band in a way that 'the Penguins' doesn't).

Generally, in the 1920s era where folk acts (very broadly defined, including jazz) first began to be recorded in serious numbers, bands have names like the Joe Bloggs Orchestra or the Texarkana Minstrels that are rather more concrete in literally describing the band rather than abstract. But you do get the Royal Troubadours in 1927 doing 'My Blue Heaven', and the Mississippi Sheiks in 1930 doing 'Bootlegger's Blues', who have elements in their name that are abstract.

However, broadly speaking, I would date the increasingly widespread use of abstract names for musical groups that led to 'The Beatles' as dating from the African-American 'barbershop' and then 'doo-wop' vocal quartets - The Ink Spots, whose first recordings under that name date from the 1930s, being particularly influential. Generally, it's thought that the trend towards avian names in African-American vocal groups starts with The Ravens, who first recorded in 1946 (and who were soon followed by The Orioles, who first recorded in 1948, and then The Crows, the Flamingos, the Penguins, etc). Before the Beatles, there was also a trend for instrumental rock bands in the late 1950s to have entirely abstract names - The Shadows, The Champs, The Virtues, the Royaltones. But it's probably also fair to say that, after the Beatles (and to a certain extent the Beach Boys), the popularity of the abstract musical group name exploded exponentially, and that there were very few 'bands' (with an abstract name, and were self-contained in terms of having both singers and instrumentalists contained within the name) before the Beatles - the obvious influence on the Beatles being The Crickets (who had a career after Buddy Holly's death, when they were unambiguously just The Crickets rather than Buddy Holly and the Crickets).

After the Beatles become very very successful, very many bands hoping to emulate their success had abstract names (e.g., The Byrds, The Doors, The Swingin' Blue Jeans, The Rolling Stones, The Knickerbockers, etc) which arrived in the wake of the Beatles' success - where the charts had previously been largely filled with named singers, with occasional vocal groups or instrumental rock bands, the bands became much more prominent on the charts. The abstract non-The rock band - e.g., Pink Floyd rather than The Pink Floyd - arrives on the charts ~1966, with the likes of Love, Harpers Bizarre and Cream being the early examples that were widely imitated in the later 1960s (along with, perhaps, earlier examples like Them, Sounds Incorporated, and Hedgehoppers Anonymous).