Why do a lot of Doris Day's song sounds a lot like Ella Fitzgerald? Were their record companies trying to pit them off? I searched it and it seems that they recorded to very different labels. Perhaps in a effort to make the audiences judge who sang which rendition better? If so, who won? I also seem to notice that Ella doesn't speak much about the strange musical relationship between her songs and Ms Day's. I know she's not much of a talkative person, but I don't see any interview of neither one of them claiming they are friends either. Was there some sort of rivalry?
In 2019, popular artists generally don't sing the same songs. There's an idea that there's a definitive version of a song, and that definitive version is, usually, the original. And why would Taylor Swift bother doing a version of, say, 'Bad Guy'? It'd be seen as just a cover, making her no better than some kid on YouTube with a ukelele. Go back 60 years to 1959 - the time of Doris Day and Ella Fitzgerald - and basically, music consumers did not feel this way about popular artists singing the same songs.
If you look into the history of the song you mention - 'Cheek To Cheek' (written by Irving Berlin), 'Stars Fell On Alabama' (Perkins/Parish), and 'Dream A Little Dream Of Me' (Andre/Schwandt/Kahn), they inevitably were written in the 1920s and 1930s by 'Tin Pan Alley' songwriters, often for the Broadway musicals or Hollywood movies of their day; 'Cheek To Cheek' came to prominence due to being featured in the 1935 Fred Astaire/Ginger Rogers movie Top Hat. The name 'Tin Pan Alley' refers to a specific street in the Flower District of Manhattan where most of the music publishers who had the rights to these songs had offices, but the music is more or less (a subsection of) the pop music of the 1920s and 1930s.
By the 1950s, after the swing era had died out, and in jazz there was a distinct decrease in tempo, and an increased emphasis on improvisation and song, there was talk of 'the Great American Songbook' or 'jazz standards' - a series of songs that were very commonly used in the jazz idiom due to their ubiquity; those songs meant that a singer or an instrumentalist could sing/play the melody and due to the song's ubiquity, the audience could pick up the themes and variations, and have a sense of how the musician was carefully playing with the melody, etc.
There's earlier examples of music that's specifically marketed as being from the Great American Songbook (Mabel Mercer's recordings in the 1940s are a very early example), but for example Ella Fitzgerald spent much of the later 1950s and early 1960s recording for Verve records, and her producer Norman Granz produced a series of Fitzgerald albums that were based around the 'songbooks' of great Tin Pan Alley composers: there's 1956's Ella Fitzgerald Sings The Cole Porter Songbook and Ella Fitzgerald Sings The Rodgers And Hart Songbook, 1957's Ella Fitzgerald Sings The Duke Ellington Songbook, 1958's Ella Fitzgerald Sings The Irving Berlin Songbook, 1959's Ella Fitzgerald Sings The George and Ira Gershwin Songbook, 1961's Ella Fitzgerald Sings The Harold Arlen Song Book and 1963's Ella Fitzgerald Sings The Jerome Kern Song Book.
Similarly, if I look at the tracklist of Frank Sinatra's 1957 album A Swingin' Affair, there's a variety of songs which Ella Fitzgerald would have done on her songbooks albums - from the Rodgers and Hart songbook, Sinatra does 'I Wish I Were In Love Again' and 'The Lady Is A Tramp'. From the Gershwins' songbook, Sinatra does 'I Got Plenty O' Nuttin' and 'Nice Work If You Can Get It'. From the Cole Porter songbook, Sinatra does 'Night And Day', 'At Long Last Love', 'You'd Be So Nice To Come Home To' and 'From This Moment On'. From the Duke Ellington songbook, Sinatra does 'I Got It Bad (And That Ain't Good)'. From the Jerome Kern songbook, 'I Won't Dance'. There's also covers of 'Stars Fell On Alabama' (a song originally from 1934), 'The Lonesome Road' (originally from 1927), 'If I Had You' (originally from 1928), 'Oh! Look At Me Now' (originally from 1941), and 'I Guess I'll Have To Change My Plan' (1929).
The Miles Davis Quintet albums recorded for Prestige in 1957 (Steamin' With The, Workin' With The, etc) also feature plenty of jazz standards - 'It Could Happen To You', 'My Funny Valentine', etc - as does Chet Baker's vocal albums of the same era, as does (etc).
Basically, in the 1950s, people did not consider that a song belonged to an individual singer in quite the same way we do now; it's also worth remembering that in the 1950s, a recording was considered exactly that - a recording of an individual performance. It was only in the 1960s that people started to see recordings as perhaps being more perfect than any individual performance, once studio technology and multi-tracking abilities advanced, and once - with rock bands like the Beatles or the Beach Boys - the specific quality of the sounds on the record became just as important as the songs. With something like 'Tomorrow Never Knows' by the Beatles or 'Good Vibrations' by the Beach Boys, you had definitive versions of the songs that were specifically designed to be heard that way on record, and which people came to see as definitive. In contrast, there was no definitive version of 'Cheek To Cheek', as far as people of the 1950s (and earlier eras with pop music and records, for that matter) were concerned - sure, it was associated with Fred Astaire, but it was everybody's song, anybody's song.
So Doris Day could do the song and Billie Holiday could do the song and Ella Fitzgerald could do the song, and there wasn't a sense that they were covering Fred Astaire - they were simply playing a song that was in the standard repertoire of a jazz-associated musician, that everybody kinda knew.