Do all Germans who went to a "Gymnasium" write in the same manner?
We-ell, not entirely. However, one does learn how to construct sentences (including verbs, subjects, objects, etc.), one learns correct spelling, and one learns ways of constructing an argument (e.g. set out the background, give an overview of the position against, the position in favour, pick apart the arguments one disagrees with, put forward one's own propositions, etc.).
That said, the same could be said for English schools, or, for that matter, for schools around the world. Not everyone will take this on board, some people will be rather better at it than others, some will add their own flair, while others will be fairly formulaic in their arguments, but still.
That does not mean that everyone writes the same way, or indeed that they are "clones" of one another who have little original thought. It does mean, that there are certain writing conventions in any language, and adherence to those (at least loose adherence) will make it easier to communicate.
Once can, of coure, deliberately break certain conventions (as Churchill famously wrote on a draft speech, where someone had criticised his use of a split infinitive "That is the sort of shit up with which I will not put"), but that tends to be done for special effect, rather than because it is accepted.
In a way, you could argue that all German posters on this forum write in the same way, because they all write in German. However, that would be missing entirely the huge variety of styles, turns of phrases, methods of argument etc. that are being employed.
And, before you ask, I did spend some years at a German "Gymnasium".
Edit: "I hereby present the Father Jack Hackett Award to LEO's very own Doris" *APPLAUSE*