An anamorphic system can be thought of as having two lenses, one for the horizontal field of view, one for the vertical. Thus the "spread" of the bokeh balls will be different for each axis, with there being more spread vertically (when behind the focal plane, as they typically are) than horizontally. This makes ovals.

Why do anamorphic lenses have oval bokeh?

Very noticeable in anamorphic shot films, oval bokeh is caused by the oval entrance pupil on the lens. This is then squeezed onto the film or sensor causing further elongation. The reason they do not unsqueeze like the main subject is because objects out of the focus plane are stretched further than objects in focus.

Why does anamorphic look cinematic?

An anamorphic lens is designed with additional glass elements that squeeze the image horizontally, allowing filmmakers to capture a wider field of view than the film or digital sensor would ordinarily allow. In production there are typically two classes of lenses — anamorphic and spherical.