It's interesting that there is so much debate on which bass Paul used on which track. The Hofner, Ric, and Jazz are all drastically different in design/construction. But it is very difficult to identify them by ear.
Paul always sounds like Paul, because the tone is in the fingers...and heart. Not so much in the instrument. And certainly not in the species of wood, the type of bridge, the nut, etc.
Macca's tone is very much about his technique, first and foremost. His use of the pick and palm muting are among the biggest factor in getting his sound.
But there's a point at which selecting a specific instrument is also going to affect the tone. The bass tone on "Yer Blues" from The White Album would be pretty tough to get with the Hofner. And did Macca have an instrument that could get that springy sound on "With a little help from my friends" with a bass other than the Rick? It's debatable. Especially since tone shaping and recording technology in the mid 1960's were limited in their capabilities.