After reading your question, I thought "I don't know, but does it have to be identified?"
In modernized folklore Japanese children hear today, it is true that there are many references to "non-descript" 神様. For example, most Japanese people know the story regarding why cats were not chosen as part of 十二支, where unidentified 神様 determines the 12 zodiac animals.
I think it was intentionally made "non-descript". There are very few people who are interested in associating it with traditional deities in specific legends or religions. As a native Japanese speaker, I personally have never wondered which religion or country those 神様 belong to, either. Japanese people today are perfectly fine with the idea of 神様 that belongs to no particular religion. FWIW, I have never heard of Jade Emperor.
When Japanese people use the word 神様 today, it perhaps refers to Christian God if they are attending a Christian-style wedding, and shinto gods if they are visiting a shinto shrine. Otherwise, it has no relationship to a particular god in traditional lore or religions. People sometimes say something like どうなるかは神様にしか分からない ("Only God knows what will happen"), in a hospital for example, but if you ask "Which God are you referring to?", you'll probably get a blank stare.
If you're interested in the serious history of 十二支 or 七夕 in China, please ask about them elsewhere. I can only explain how the word 神様 is perceived by ordinary Japanese people. Of course there are well-known legends that are unequivocally tied to shinto gods, but they are generally not popular with kids.
Maybe related: Japan: The Most Religious Atheist Country
EDIT: Looks like the one called 神様 in so-called the "Tanabata story" can be traced back to 王母娘娘. I believe very few Japanese people know this. As for 十二支, some sources suggest this 神様 is お釈迦様, but the history of 十二支 is longer than the history of Buddhism itself...