After a bit of research, I think both terms are good here.
Mathworld says for the definition of 'normal vector':
"The normal vector, often simply called the "normal," to a surface is a vector perpendicular to it."
So it makes sense to talk about a vector perpendicular to a surface (we knew that of course).
'Normal' is just a more sophisticated way of saying the same thing.
And it doesn't seem right to say that 'normal' and 'perpendicular' are synonim, because the term normal, as I understand it, can only be used when talking about a vector. So we can say that two planes are perpendicular to one another, but we cannot say that they are normal to one another.
I would say 'no'. 'Normal' is only used to describe perpendicularity of a vector wrt a plane.
Have you seen this used in a textbook before? I think they mostly use the term 'orthogonal', which is a synomim to perpendicular when talking about simple object such as lines, vectors, planes, etc., but actually extend to more abstract mathematical objects as the generalisation of the notion of perpendicularity.