By pretending her last name is pronouced "Bouquet" instead of Bucket, Hyacinth is pretending to at least landed gentry.
There are many, many family names in Britian, mostly belonging to the landed gentry, nobilty and other great families that are not pronouced the way they are spelled. This is why the show is such a hit in GB. Not saying one has to be British to understand the nuance, but it does help.
And yes, many of the aforementioned names are French in orign, which goes to another bit of social pretension. Many old familes can trace their lines back to when the Normans (French) occupied parts of what is now Britian. Those familes form much of what is known as the "landed gentry" and even some peers. So by insisting her name is French, Hyacinth is trying to establish a connection (at least socially), to those families.
Hyacinth's pretension is funny because not truly coming from such a background, the woman has no idea how things are done. Rather, she does things as she thinks those she is trying to emulate.
There was, well until Labour governments took over, a British society where everyone knew their place, and "bounders" were looked upon with distain by all. Although this supposedly has changed, to an extent it hasn't totally gone away. Hyacinth's constant name dropping, mentioning of her wealthy sister (and her possessions) would NEVER be done by those truly born to the manor, that is what makes her so funny. It also give her away every time in front of true society. They know at once she is "not one of us", and humor her only because it is impolite to be rude. Everyone else either is dragooned into service, or just cannot get a word in edgewise to shut the woman up.
L.