A friend of mine has a Höfner Violin Bass from 1965.
(as a German it is easyer to type the "ö" because it's on my keyboard
You can't really compare a cheap Höfner or even the expensive new Höfners to the 1960s Violin basses. The 1960s sound more "open" / "airy" / "wooden" ...
The finish also looks different because the 60s had a nitro finish, so an original bass looks semi gloss / semi-mat.
It was surely called "Beatles Bass" because the Beatles were known all over the Western (and Eastern) world. The Beatlemania was in Germany too - and what better could happen to a German instrument builder than the Beatles, playing their instruments?!
The audience in Hamburg surely remembered Stu, but even the majority of "teen fans" from 1962/63 on only knew Paul as the bass player. But Stu isn't forgotten!
And also not Klaus Voormann, bass player himself and an old friend of the Beatles since the Hamburg times (and creator of the "Revolver" cover). As far as I know, he's still Paul's friend and sometimes plays with him ...
You call it "Beatle Bass"???
In Germany it is called "Beatles Bass" ...