Do the length scale and velocity have to be orthogonal in Re
 

Does a Reynolds Number always have to be defined with a length scale and velocity that are orthogonal to one another? Could the length scale and velocity be parallel?

Any choice is possible and produces a non-dimensional Reynolds number.

You have to understand the proper couple (velocity, lenght) that characterizes your specific flow problem.
Just an example is the Reynolds number for an airfoil, you use the free stream velocity and the chord.

But on a channel flow you use the height.

In a correct approach to any problem, you don't simply choose a Re number definition, you use the Buckingham Pi theorem.

You still pick a length and velocity scale for that, but if others exist then your problem is not completely defined only by your Re, but also from all the other length and velocity ratios.

Let's take the airfoil example. The far field velocity is the only velocity scale, but as length you have the chord c and the thickness t. Whatever one you choose, your problem will be still defined by two nondimensional numbers: Re and t/c.

If you are smart enough, you can pick up the correct ones in each non dimensional number so that your quantities of interest will show clear asymptotics with nondimensional numbers going to 0 or infinite. Otherwise, the correlations you will find will, most of the times, clearly show you what is wrong. When they don't, it's probable that the problem itself has no such clear influence from one single parameter.

Also note that the single/simple Re number dependence of a flow is, in the end, a very simplified frame that really assumes no length and a single velocity scale are present in the inflow. A real turbulent inflow would have a continuous distribution of them.

A similar reasoning is possible for the actual surface of the airfoil, whose roughness also would have a continious distribution of length scales.

This just to say that the Re number and similar quantities are just numbers and are not written in the stone. If you need to use an existing relation involving Re, you obviously need to use the same Re definition if you want to correctly use the relation. Otherwise, your physical intuition should come first of even attempting to come up with a dimensionless number of any sort.