If a matrix $P$ is such $P^{-1}MP$ is diagonal, we say that $P$ diagonalizes $M$ (implicitly, as the matrix of an endomorphism).
Now, if $P^\top M P$ is diagonal, is it correct to say that $P$ diagonalizes $M$ as (the matrix of) a bilinear form? It seems correct to me but I have never read this, so I am wondering.
On
The natural operation for changing variables in a quadratic form amounts to the expression $P^T H P,$ where $H$ is the Hessian matrix of second partials of the form, and $P$ is nonsingular. In matrix theory, $H$ and $P^T H P $ are called congruent. There is an evident equivalence relation.
Sometimes the Hessian, or half the Hessian, is called the Gram matrix of the form.
When $\det P = 1,$ then $P^T H P$ and $H$ define quadratic forms that are called equivalent. In dimension two, care is taken, but in higher dimension $\det P = \pm 1$ is generally used.
If all entries of $H$ are integers and all entries of $P$ are integers, the revised form is called equivalent; if all entries of $P$ are rational, $\det P = \pm 1,$ then the forms of $H$ and $P^T HP$ are in the same genus, although there is a condition called "without essential denominator." This definition is due to Siegel.
Maybe an example. Let $f(x,y) = x^2 + y^2.$ The Gram matrix $G$ is just the identity. Then $$f(3x+y, -5x + y) = 34 x^2 -4xy + 2 y^2.$$ Well, $$ \left( \begin{array}{rr} 3 & -5 \\ 1 & 1 \end{array} \right) \left( \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array}{rr} 3 & 1 \\ -5 & 1 \end{array} \right) = \left( \begin{array}{rr} 34 & -2 \\ -2 & 2 \end{array} \right) $$
Yes, sure that is perfectly normal terminology (for example here, or here, or here or here. Frankly I'm a little surprised you never ran across this usage.)
The thing in common between the two processes is that they take two matrices that are expressing something as a matrix (resp. linear transformation, bilinear form) and connecting it with the new matrix after a change of basis (using, resp. similarity, cogredience). Whichever equivalence relation you are using, it makes perfect sense to call the classes represented by diagonal matrices "diagonalizable."