Looking at the equation, which is written all over my textbooks, $f=f(x)$ or $\mathbf r = \mathbf r(x,y,z)$ or what-have-you, I can't help but think that that is just wrong. On the LHS is a function. On the RHS is the value of that function evaluated at some arbitrary point in the domain. I understand that what it's supposed to mean is something like, "$f$ is a function of one argument which we'll be calling $x$", however, it looks to me like it is mathematically unsound because there are different types of objects on the LHS and RHS.
Is my interpretation correct? And if so, why do we use this notation?
This notation tells you that the function will be called by two different names in the sequel. When the argument is important, it will be called $f(x)$. However when the argument is not particularly important, it will also be called $f$.
In this expression, the symbol $=$ is not used mathematically, but meta-mathematically, as shorthand for "which we will also call".