The figure of speech "in this vein" can be viewed as a single branch within a larger system. In addition to "manner and mode" some other synonyms are: attitude, streak, temperament, characteristic, way, style, disposition ...

It can also mean "in this mood". In a comic vein = in a funny, comic mood. For 'style', you would normally be referring to literature. "She writes in a romantic vein." = in a romantic style. Other synonyms are manner, mode.

A vein is a channel which guides the flow of the blood. Likewise it can guide the flow of the style.