Alright. I didn't have the time earlier and figured that another Russian speaker might bounce in and help you out. I saw you got a partial answer in /r/conlangs. I can look through that book for you and look for any answers to questions you may have, if you still want it.

As /u/Qwatuwayh pointed out, there are actually several scripts on the website you linked and they are indeed a hypothetical reconstruction based on early scripts found by archaeologists in that region. Toward the beginning of the book, I found the following passage:

Силлабо-пиктограммы Махошкушха (Северо-Западный Кавказ, современная Адыгея), датируемые 9-8 тысячелетием до н. э., являются, по всей видимости, древним письмом северокавказцев, к тому времени уже выделившихся из сино-кавказской макросемьи…

Syllabo-pictograms found near the mountain of Makhoshkushkha (Northwest Caucasus, present-day Adygia), dated to the 9–8th millennium BCE, that seem to have been the ancient writing of the North Caucasians, who by that time had already split off from the Sino-Caucasian macrofamily of languages.

Translation mine. NB: "macrofamily" is a technical term for Russian linguistics meaning basically a large language family or superfamily. Likewise, Sino-Caucasian is a hypothetical language proposed by Sergei Starostin, a major Russian linguist. In Anglosphere linguistics an analogue seems to be Dené-Caucasian.

Let me know if you have further questions about what's in that book you linked.