Relative flow over a lifting body has to be described as a vector because Lift and Drag are defined wrt the relative flow vector, and Angle of Attack is a parameter.
Is the sum of squared velocity vector components referred to as velocity squared, in aerodynamics?
Yes. Velocity is used to refer to the vector as well as to it's scalar magnitude in scalar formula. This isn't sloppy, it's compact, and more importantly, it maintains the connection to the underlying vector domain.
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The squaring-something operator actually represents a large number of different algorithms that are specific to the thing being squared.
Sometimes, there is more than one way to square a particular thing. For instance, a (scalar) distance squared might represent an area, or it might not. But the same symbol is used for the operation, and the associated units are often rendered identically. Sometimes it is important to keep track of the distinction and sometimes it isn't. An area has properties that the square of a distance does not have. Just because two things have the same units doesn't mean they have the same set of properties.
The same applies to velocities and speeds. Both speed and velocity have the same units, but they have different properties. Both can be squared, but the procedures are different.
So to get square of the speed of the relative flow, you take the magnitude of the relative flow, then square it. To get the square of the velocity of the relative flow, you square the velocity (a vector operation), and then take the magnitude if you want a scalar.
It's not luck that $ scalar square ( magnitude (vector))$ and $ magnitude (vector square ( vector)$ have the same numeric value and units, The operations were designed to work like that.
But do they have the same properties? Not really. When talking about velocity as a scalar, we are reminded that the domain is a vector domain and that different vectors can have the same magnitude.
Which brings us to a problem, different relative flow vectors with the same magnitude do not have the same Lift and Drag as each other. The only way the formula is valid is if the angle between the Lift and inflow remains fixed and the angle between the drag and the inflow remain fixed, which they do by definition. But all these have to remain fixed with respect to the lifting body as well. So the statement is only valid for constant angles of attack. To suddenly introduce a vector constraint on an entirely scalar chain of math is plain bad form.