Can one linearise a function of matrices in a similar way to a scaler function
i.e $$f(x) = (x^{-1} + c^{-1}) \approx f_a(x) = (c + a)^{-2}\left(c^2x + a^2c\right)$$
(Where $f_a(x) $ is the the approximation of $f(x)$ as a straight line at $x= a$)
Does it still make sense to do something like the above if x is a matrix?
$$f(X) = (X^{-1} + C^{-1}) \approx f_a(x) = (C + A)^{-1}(C + A)^{-1}\left(CCX + AAC\right)?$$
Depends what you mean by "makes sense".
At the very least, you'd have to be more careful with the order of products, because matrices do not commute. But in general the picture is more complicated than that, because you can perturb a (square) matrix along $n^2$ different directions, not only one: the correct generalization of linear approximations to matrix spaces is the Fréchet derivative, which is an operator from $\mathbb{R}^{n \times n}$ to $\mathbb{R}^{n \times n}$.
In your case the function is essentially $X \mapsto X^{-1}$, so its Fréchet derivative is well known (see e.g. here).