Is there any reference(for ex. wears Janoi) in any Vedas or in any other book that describes God as brahmin (by caste)?
One reference I found in Hanuman Chalisa is,
हाथ बज्र औ ध्वजा बिराजै ।
काँधे मूँज जनेउ साजै ॥५॥
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All male Gods(Ganapaty,Shiva,Vishnu etc) wear the Yajnopavita.And we can't arrive at any conclusion regarding the caste(varna) of them just by looking at their Upavita.
For example,Ketu(one of the Navagrahas) is regarded as an Antyaja(baseborn/outcaste) in the Brihat Parashara Hora Shastra.But as per the Ketu Pancha Vimsati Nama Stotram(The Stotram comprised of the 25 names of Ketu),found in the Skanda Purana,even Ketu wears the sacred thread.
raudrō rudrapriyō rudraḥ krūrakarmā sugandhadhṛk| palāladhūmasaṅkāśaścitrayajñōpavītadhṛk||2||
He who is angry, he who likes Shiva, he who is fierce, He who does cruel deeds, He who is seen by pleasant smell, One who resembles the smoke of husk, one who is seen wearing sacred thread.
So,the rules that apply to the human beings with respect to wearing the sacred thread(that is the Dvijas only should wear it) may not be valid for the Devas(divine beings) as well.