In classic Warner Bros. cartoons, it wasn't uncommon to see a specific set of major faces from the period, mostly with girls swooning over 'Frankie,' but Bing Crosby, Al Jolson, Charlie Chaplin, and Bob Hope seem to have appeared in them with some frequency. These often included parodies of their acts. One would occasionally see other 'cameos' of sorts with Cary Grant and others, but those five seem to have been the most prevalent.
Fast forward to the era of Hanna-Barbera and even the revival of Warner Bros. animation, and with the exception of special appearances (e.g., the Globetrotters), you don't see much in the way of contemporary faces except for the rare, mostly static shot.
Is this due to a change in image control, or were Sinatra and the others such oversized stars that no one else ever matched up?
One major motivation behind the classic Warner Brothers cartoons was that, with the introduction of film sound, Warner Brothers became aware that they needed songs and music for their films.
However, in order to use copyrighted songs in their films and not get sued, they needed permission from the music publishing companies that published the sheet music and thus controlled the copyright associated with the text of the music. As a result, two years after the enormous success of 1927's The Jazz Singer (the Warner Brothers film starring Al Jolson which famously was one of the first to use sound), Warner Bros. simply bought up several prominent publishing companies so they could use their music.
This is the reason that the Looney Tunes and Merrie Melodies series have musical titles: these cartoon series were originally envisaged primarily as ways to exploit the catalogue of songs in their music publishing business. As a result, music was very prominently used in many Warner Brothers cartoons, whether it was Bugs Bunny singing opera, or a talking frog singing "hello my baby...". And where these cartoons used copyrighted music, they invariably used songs that were currently in Warner Brothers' music publishing catalogue. And if the Merrie Melodies and Looney Tunes cartoons used songs in the Warner Brothers Music Publishing catalogue that were heavily associated with particular acts - Al Jolson or Bing Crosby or Frank Sinatra - it made sense for the cartoons to acknowledge this by way of parody or references.
It's also worth mentioning that, in the days before the widespread adoption of television in the US in the 1950s, the Warner Brothers cartoons would have largely been seen in cinemas as a short film to be played before a main feature film. Take the 1946 Merries Melodies short 'Bacall To Arms', which parodied the Humphrey Bogart/Lauren Bacall films of the time (specifically 1944's To Have And Have Not). I can't find any evidence of this, but one suspects 'Bacall To Arms' might have gotten played in some cinemas as a short before the 1947 Bogart/Bacall feature film Dark Passage. In any case, the Warner Brothers cartoons were being played to movie audiences, and so it's unsurprising that many of them use references to film stars of the time such as Bob Hope or Cary Grant.