Why and how personal philosophies of certain thinkers (starting from Ancient Greek ones and up to modern ones) became recognized as "classical" main stream approaches ?
For the very simple reason that they have been passed on by subsequent thinkers, while all others have not.
It might not seem to you like a satisfactory answer, but the following question why have certain people's ideas been passed on and others not? has no unique answer, and explanations are as varied as historical explanations can get: All manuscripts but a few burned in a certain place. A certain religious background allowed some ideas to thrive and others where lost into oblivion. A new science becomes successful and some obscure ideas on which the scientific endeavor is based suddenly become common sense. Not to mention the creation of "classics" out of thin air by backdating later ideas to previous thinkers to forge arguments from authority.
In this sense it can be said that what makes up the canon are no "personal philosophies", but highly adapted ideas to a certain point in time. Also, your question implies that what makes up a philosophical canon simply accumulates over time. The most important lesson here, I think, is that there is no such static mainstream, an overarching canon. "Classics" change from century to century.
I don't think most have ever heard of Eugen Rosenstock-Huessy or Hugo Dingler, an opportunistic Nazi-philosopher? During their time they were very popular (and sometime influential), today they are forgotten and will likely not survive another century. Yet, e.g. Dingler's weird concept of science survived in the work of his pupil Paul Lorenzen and got us the "Erlangen program", a rejection of relativity theory and the creation of game semantics. (You can guess which outcomes have become popular and which have not.)
Doing research in the history of philosophy can be a daunting task at times for exactly this reason: e.g. you might want to understand certain nowadays "classics" in their own time and place and discover - not without puzzlement - that you have to acquaint yourself with obscure figures and ideas in order to proceed any further. Sometimes these figures were much more popular during their time than today's "classics".
A nice thought-experiment is to try to imagine what will be passed on from 20th century philosophy in, say, 200 years from now. Inevitably one will come up with a list of people popular and seen as "on the right track" today. If history can give us any guidance, the criteria for the inevitable selection process will be established by what will be deemed interesting in 200 years, most certainly not what has been popular and deemed "the right direction" in the 20th century.
"Mainstream" is a vague term. Let's say that by "mainstream philosophy" you mean the major theoretical positions held by large numbers of professional philosophers. In the english speaking world most professional philosophers don't think Plato is right about anything. They don't think the soul is immortal, or that all knowledge is by recollection or whatever.
Mostly, mainstream philosophers argue about things like: "If somebody knows that p, do they know that they know p?" or "Do we have a priori knowledge?" Some of the problems that philosophers today are worried about are old problems, and in the contemporary state of the art, there are in fact positions that are sometimes similar to positions that someone held in the history of philosophy. For instance, some ethicists today espouse approaches to ethics that are similar in some respects to Aristotle and Plato's. However, the reason to hold those positions has nothing to do with the authority of Aristotle. The reason the position is mainstream is that it has a good argument in its favor, or it solves some otherwise insoluble problem. Philosophy as a discipline makes progress just like other disciplines do in that bad ideas tend fall out of the mainstream over time as they are shown incorrect (at least in theory).
I think your question seems to be supposing that Socrates's "personal philosophy" was just a set of thinks Socrates happened to think. But that's not true. Socrates's philosophy grasped his contemporaries, and still grasps modern readers, because of the arguments he offers for this claims. On further philosophical reflection, those arguments aren't as strong as they first appear, of course. But that doesn't mean Socrates was just making it all up. At the very least, he got the ball rolling for us and so the fact that the discipline has left him behind is, in a way, a tribute to his success despite the falsity of his philosophical beliefs.
It strikes me as odd that none of the suggested answers have yet discussed the Renaissance. To draw a brush stroke very broadly, the reformation in Europe prompted a decline in the singular intellectual authority of the Christian Catholic Church, at which point scholarship in texts from antiquity, the European Roman and Greek civilisations that immediately pre-dated the spread of Roman Christendom, found renewed interest. These pieces had already some influence through their use in exploring and developing the Scholastic tradition of philosophy in Christian Theology, and by re-exploring these outwith the need to establish Christian canonicity, new ways of developing models and exploring concepts of good living emerged.
"Classical", in that tradition, simply refers to the Greco-roman poets and philosophers as influential and cultural sources (regionally local to the Civilized World of Europe) from before Christianity. If one engages in one's own work with the tradition that followed from that Classical rediscovery, then this too can be said to contribute to a Classical discussion