The movie takes place in 1910. There are two musical numbers that I am wondering if they are accurate to the period.
First is the introduction of Thomas O'mally the alley cat. He sings a song that I would assume is more of a big band style that I would associate with the 40s or 50s. https://youtu.be/yRET1vsfiJM
Second is the jazz piece Everybody Wants to be a Cat. Cat seems to have a double meaning of a literal kitty cat and the Jive meaning of a being a cool cat or jazzy person (excuse that terrible definition). Jazz has a long history and this piece seems more modern, but I'm not sure. They use the word groovey which I dont think was popularized until the 20s. https://youtu.be/4rrXR6n0RTY
Firstly, yes, ‘jazz’ as a genre did not really exist in 1910; the genre being named as such comes later in the decade. The musicians in New Orleans playing a homegrown version of the ragtime style which later developed into ‘Dixieland’ style jazz would not have understood themselves to be playing ‘jazz’, and the music would not have sounded like the Thomas O’Mally introduction; it was generally instrumental dance music in its early years with a focus on sounding somewhat chaotic, initially often with instruments imitating animals and sirens and such things.
The singing style of the voice actor playing Thomas O’Mally is quite clearly a post-Bing Crosby crooning vocal style, a style that developed in the 1920s as recording techniques and amplification techniques improved. You can hear a 1928 track by Paul Whiteman and orchestra featuring Bing Crosby on vocals here which is closer to the style used by O’Mally; 1928 was Crosby’s breakout year as a pop star.
According to etymonline, the idea of ‘cat’ as a cool person dates to 1920, and the jazz connotations date from 1931.