I usually associate class with social connections, standard of living, educational achievement, and occupation, and financial security usually follows all four, but not necessarily, e.g. someone who merely marries rich.
Yes and no: a class is not just a isolated, self-sufficient ecosystem, built on certain relations (those connections you mentioned) and common interests, and habits (the right places to attend, whether for leisure or education... social interaction in general), but also THAT in a global perspective in which every class entertains a definite role with respect to all the rest: otherwise, what you described is merely a particular group, tribe, whatever people with certain habits and interests. A social group is so precisely for the social part, for BEING, for existing in relation to at least another different group.
"Standard of living" and "financial security", even the connections and the rest only make sense to a particular dynamics that you take for granted while being the heart of the matter, and what you mentioned are impressions from the XIXth century bourgeois system, for you merely American system, but which is a system that I describe better in the following paragraph.
This is nothing of a belamian effort: we all learned at school that the class system was a "minimal improvement" on the old feudal estates of the realm, allowing a certain permeability impossible in the old system that was tied to a natural vision of the world in which each being WAS a certain moral and, therefore, social condition transmitted by birth.
Maybe that's confusing for the American people who live in a nation which was built precisely on severing itself from the burden of that past (although the history of America in the past two centuries shows a definite and ruthless effort to remake the European old system, for all the enthusiastic rhetoric of freedom and equality and opportunity through personal achievement), but the facts are there for anyone to see below the prejudices or assumptions.
So, the reason why there are no classes today is that they are not necessary anymore in a system in which wealth do not even need people, even qualified people: the problem with the perception of the state of things is that people expect reality to be unified, simple, while in fact eras show what appear as a duality: some idiots may relate this to Marxist dynamics, but that dynamics was as static as the instinctive unitary one, with a duality of static states, while what I am talking is of a mere state in progress not of two opposing natures, as Marxism proposed, but two (at least two main ones) manifestations of a single principles, evolving one from and through the other.
Tocqueville wrote about an obvious tendency to democratization in the Western world from the Low Middle Ages, but it is not actually so, and the 1%-99% thing is a proof of that: the tendency has been the progressive replacement of one source of wealth by the other, achieved through technical development, all of which needed to do without any sort of class boundaries: that is the "Western specificity" that Western theorists, puzzled, expected to naturally arise from specific conditions that they observed in other civilizations, mainly in whatever was called "China" in different eras.
So, class system is only an instrument to maintain a certain order, in order to achieve certain welfare goals, your "standard of living" and "financial security"... Cod I am getting bored writing this, but I have no time to truly develop and, at the same time, need to shout it ... what must be taken for granted is that only one small fraction of the people will, for whatever reason and in whatever way, actually opt and operate to establish themselves as that fraction, and the American dream is not about everyone being rich, not even everyone being given the same opportunity to be a great achiever: it is not about America, in fact, but about those European hindrances I referred to above, which are also the norm everywhere else, in any other era.
The American dream and the social system derived from it is: be gutsy enough to impose yourself to all the other tamer ones... if everyone were as greedy as Gordon Gekko, there would no possibility of greed at all.
Once social divisions are not indispensable to be "rich", to establish yourself as one of the happy few, once wealth comes through algorithms without the need of there even being anything "actual", a product to which to assign value, you can also dispense with classes of people.
You obviously still need a set of qualified technicians to operate that system, who do not need to belong to that upper class just like they never worked in their own land in the flourishing, ripe old system, and that is where that "duality" of systems, with a "class" with a definite role has a place-
Finally, the fact that second generation upstarts are accepted as one of the group more as a norm than as an exception, shows how slovenly and irrelevant the old system has become: formerly classes were about generations, today, in accordance with all said above, the "bourgeois" morals of achievement, today it is about individuals, individuals being shaped in their minds and whole vital experience according to a definite perspective of your "financial security" and "standard of living", of my "habits" and both your and my "connections" that you can possibly replace through personal effort and achievement... one belongs to a class when one belongs to effortlessly. That's still the old system at work. The other newer system is what we see so often at work in Wall Street: socialism for the few, based on greater relative wealth.
I know this is a big piece of TLDR, but I had to spew it, but I am positive some very few, weasly ones could profit from it.