I would glue them. If it has never been glued and there is raw wood in the joint then any wood glue would work fine. If it has been glued before or the wood is sealed in any way I would use a slow set two part epoxy.
If there was glue on the dowels then I would re-glue them together.
I have seen chairs made with no glue at all. They were designed that way and the type of wood & moisture content selected for the chair was critical in order to avoid glue.
Looking at the photograph I doubt your chair was intended to be made without glue.
There is the possibility the chair was not glued for shipping purposes. I did some work for an antique dealer that was buying furniture from Taiwan which was completely finished like that but it was unassembled and neatly packed in boxes. You could fit three or four of these chairs in a suitcase the way they had them packed. These chairs could easily be assembled without any glue however the antique dealer was selling most of these chairs to restaurants and wanted them more durable so he had me permanently assemble them.
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