This mostly came up because I saw that some of the old British cavalry units became armored cavalry and still serve today. Was service in these units sought after by upper-class people to gain social standing and honour, as well as becoming traditional for some families (or even maintaining tradition from the old cavalry days)? If not, then what caused this discontinuity between cavalry and its modern equivalent?
It depended on the unit. As you probably know, some units (at least in the UK) were considered to be more "fashionable" than others. So if you were a blue-blooded aristocrat who wanted to lead an armored unit odds are you'd try for something like the Guards Armored Division (essentially foot guards in tanks) or perhaps the Life Guards. Or a unit like 7th (Queen's Own) Hussars might be more desirable than local yeomanry unit. 1st Royal Tank Regiment, as a newer unit, might not have the same allure.
As for continuity between horse and tank, some tankers certainly saw themselves as cavalrymen in armored steeds. George Patton certianly And some tanks were built to act in cavalry-like ways. British Cruiser tanks were meant to carry out the age-old cavalry responsibility of exploiting breakthroughs and pursuing retreating foes.
But not every tanker saw themselves as a modern-day cavalryman. You certainly wouldn't see Percy Hobart toting a riding crop