If the flat plate were not there, the flowfield would be perfectly uniform and could be represented mathematically by a velocity potential increasing linearly in the flow direction. Such a potential flow would be inviscid, and if you were to define a Reynolds number for it, you would have to consider it to be infinite.
Introducing the flat plate parallel to the flow results in a thin layer close to the plate where the viscous effects are confined. It also gives you a characteristic constant length which is useful for defining an overall Reynolds number for the entire flow. (Note that there are other commonly used Reynolds numbers which are not constant, based on other lengths in the boundary layer: distance along the plate, momentum thickness, etc.)
Flows around other streamlined shapes (airfoils, for example) can be described in a similar way, with all the viscous effects occurring in a thin boundary layer near the surface. In these cases, the outer flow is no longer constant and there are velocity gradients, but the flow is still inviscid and can be represented by a velocity potential distribution.
Even in shapes where the thin boundary layer approximation breaks down, it is often useful computationally to divide the flow into a viscous region near the boundary and an inviscid region farther away.
In all these cases, a single Reynolds number is typically used for the whole flowfield, without distinguishing between the viscous and the inviscid regions.