Not true. We have forty-five more-or-less-intact scripts of ancient Greek plays, plus chunks of many other plays (very large chunks, in some cases), and while there are certainly recognisable common elements of the four main genres we see (tragedy, satyr play, Old Comedy, and New Comedy), it would be absurdly reductive to call them just a "few plays...tweaked every performance". They all have distinct plots, characters, scripts, etc.

As a side-note, it's worth observing that most of these plays -- the ones dating before 400 BCE or so, which is all but four of them -- come from a time when a play would normally only have one run (where a run equates to one performance in each centre where plays were performed). This is because they were pageants for a religious festival as much as they were dramas in the modern sense. People could buy copies of the scripts fairly easily, but full-scale performance only occurred as entries in a competition that formed part of a religious festival. (An exception was made in the late 400s, when Aeschylus' plays were re-staged; but that was decades after his death, and it was very much an exception.) In the latter 400s and the 300s, Attic drama received a much wider audience, the genre was adopted in other parts of the Greek world, and classic plays were performed on command in many parts of the Mediterranean; that's when things shifted to a model that looks a bit more like modern drama.

The only genre where your source comes within earshot of the truth is in New Comedy, which for us is represented by Menander's plays, and by Plautus' and Terence's re-makes of Greek plays for Roman audiences (the originals are all lost, except for one passage in Plautus' Bacchides). That is because New Comedy characteristically made heavy use of stock elements: stock characters, stock scenes, and to some extent stock plots. Standardised masks were used for stock characters like the Cunning Slave, the Youth, the Old Man, the Braggart Soldier, the Cook, and so on; set-piece scenes were comic highlights of the play, such as the Running Slave scene, or the Baggage Train scene; regular staging conventions were observed, like the classic setting of two houses, one on either side of a shrine, or the emergence of the five-act form; and there were common plot elements like a young man falling in love with the neighbour's daughter or with a courtesan, or a brother and sister being separated at birth, or a family member being kidnapped by pirates and miraculously being revealed as having been there all along near the end of the play.

(If you're curious, the 1966 film A Funny Thing Happened on the Way to the Forum displays many of the stock characters and stock plot elements in a very effective way; it's ostensibly based on Plautus, so the setting is Roman, but you'll recognise all of the stock elements if you've read only Menander. If you need further encouragment, it also has music by Stephen Sondheim; Zero Mostel's greatest film role; Michael Crawford and Michael Hordern in important supporting roles; and Buster Keaton's last screen performance. It does not, however, attempt to imitate New Comedy's use of stock scenes -- unless you count the chariot race at the end, but that's a Hollywood stock scene, not a Roman comedy stock scene.)

But even in New Comedy the description you quote is unfair. Stock elements do not equate to every play being the same, only tweaked a little. Greek New Comedies were different enough from one another that one of Terence's claims to fame is his talent for taking two separate plays and blending the best elements of them into a single play (a process that modern critics call contaminatio). It's a bit like saying that there's only one sitcom in existence, tweaked a little every time: an easy cheap shot for someone who dislikes the genre, but not really true if you take it seriously.