So there is not one uniquely qualified Reynolds number: every different geometry has a different range of behaviors as they scale with different parameters, and they all define different Reynolds numbers. Sometimes it is even hard to figure out with the simplest Reynolds number tables, whether the characteristic length scale should be a diameter or radius.
A typical Reynolds number application is "fluid flow around an infinite cylinder" or "the drag force on a sphere moving through a fluid" or so," which have a range of symmetries associated. In these cases $u$ is actually $|\vec v|$ and the symmetries mean that you don't need to incorporate the direction of the flow.
If your object were asymmetric, you would probably still define the $u$ as the $|\vec v|$ of the flow out at infinity, and you'd probably incorporate its direction by specifying a set of angles which would define how the object was oriented in the context of the flow. That's not the only way you can define a Reynolds number -- but there is no "only way" to define a Reynolds number.
It may help to remember the abstract point of the Reynolds number. For example I might be trying to figure out whether it makes a pickup truck more aerodynamic to have the tailgate up or down; it would be very hard to get a force gauge that can pull a truck and then drag it behind a car with a long steel cable (so that the car ahead can get out of the truck's headwind) and drag it down a long open track with the force gauge in the cable telling me what drag force was actually happening. I'd prefer if I could do all of this with a normal-sized force gauge and a long bit of string in a desktop wind tunnel, I can buy a scale replica model pickup truck for relatively cheap, and I need to know how fast is a realistic speed for that wind.
Now the equations come and help me out because I can nondimensionalize the Navier-Stokes equations, so that my entire fluid flow problem comes down to a bunch of abstract unitless numbers, with the resulting numbers from the equations giving me the correct forces "in the appropriate units" (whatever those are). So if two different systems have the same values for all of these dimensionless constants then the abstract-unitless solutions are the same, and the scale model perfectly replicates the underlying physics even though the forces and flows and everything else might be wildly different between the two.
So I choose "hey, I'm going to measure the distance from the center of one wheel to the center of the other as my characteristic length scale." I could have chosen a bunch of length scales, but I chose this one. Maybe instead of a wind tunnel on my desk I decide to pull the model truck along the bottom of a swimming pool with the force gauge, maybe that's helpful... whatever the fluid is, will set $\mu/\rho.$ The Reynolds number is then about figuring out what $v$ of the toy truck is equivalent to what $v$ for the full-sized pickup, and what velocity do I need to pull this thing at to get an understanding of what's happening at highway speeds. The velocities which correspond will have the same Reynolds numbers, which will make certain aspects of their fluid-flow characteristics equivalent. Define it however you want; it's just about getting the dimensional scaling right when you scale the model up or down.