Is teaching how to write a particular language, its character formation etc. a part of language teaching as well or not? From a language teaching/learning perspective, is the writing system a part of language teaching or acquisition as well?
If the language uses a logographic writing system, than teaching how to write is definitely part of language teaching. However, when and how the writing system is introduced depends on the teaching approach.
When learning to write Chinese characters or hanzi, Japanese kanji and Korean Hangul, for example, you will also learn stroke order. Stroke order is not only relevant to writing but also to reading: when hanzi, kanji or hangul are written rather quickly, you often can't make sense of them if you don't know the proper stroke order. (The video Is Stroke Order Actually Important? | Korean FAQ illustrates this issue.) Teachers do not always teach stroke order explicitly; they may also introduce the basics of the topic and then refer learners to relevant learning materials. There are books that actually illustrate the stroke order for frequent characters. There are also rules for the stroke order, but these are not always taught explicitly; you can pick them up as you learn to write more characters.
When you learn hanzi or kanji, you also learn a bit about the components that make up the characters, e.g. components that refer to water, fire, plants, the hand, the dog, etc. In Chinese, these components can often be used as a hint for the meaning of half-forgotten characters (though normally not for new ones). Teachers sometimes point out the components that make up a character and what they mean. When looking up characters in a printed dictionary (which some people still do), knowing about the components and the number of strokes is essential.
Some language schools and universities teach hanzi or kanji from the start, while some of them start later. As I wrote in an other answer, the sinologist Victor Mair wrote that he "would not teach students a single Chinese character until they were relatively fluent — about two years".
However, if the learner's native language and the target language share the same writing system, there is no need to teach the writing system. That is my experience with being taught French, Latin, English, German and Spanish at school or at university.
In conclusion, teaching the writing system is definitely part of language teaching when the target language uses a logographic writing system. One reason is that not being able to read or write a language make your language skills less valuable in a professional context. Another reason is that you hit a ceiling in the level of learning materials that are available.