Why does pouring tea to a plate make directional jets?

I think the elongation is caused because some of fluid (at the top) flows freely from the cup, whereas some is subject to viscous drag from the cup when pouring, and thus has lower horizontal velocity as it leaves the cup.

When the fluid hits the plate, fluid tends to flow away at right angles from the surface of the downward flow --- a perfectly cylindrical downward flow would produce a radial flow outwards from the point it hits the plate. Two such cylindrical flows next to each other would each prevent radial flow in the direction of the other. This would redirect fluid from both so that it goes perpendicular to the line between the flows.

That's easy. Notice that the jets form between the two large vertical "pipelines" that feed the intersection with the plate. Both pipeline columns create a high stagnation pressure where they contact the plate and that pressure accelerates flow along the plate in a laminar film, outward from the foot of the pipeline. Surface tension keeps the laminar flow area thin in the vertical direction. The laminar flow of both columns intersect however, causing a further, but slight increase of (dynamic) pressure, pushing the laminar flow in at that intersection both a little upward and outward in the geometry that looks like a jet. The jet flow is confined to itself because as it moves along, the intersection line of the two laminar spreading flows keeps feeding it. Turbulence eventually dominates and the more ordered flow gets broken up.