I'm learning matrix calculus and in 1 of the examples I found the following equality: $$ - \mathbf{y} \cdot (\mathrm{d}\mathbf{a} - \mathbf{z} \cdot \mathrm{d}\mathbf{a}) = (\mathbf{z} - \mathbf{y}) \cdot \mathrm{d}\mathbf{a} $$ where $\cdot$ refers to Frobenius product and all the matrices are of $m \times c$ size.
I do not understand how the author derived this. Is Frobenius product not distributive wrt addition?
Yes, inner products (and almost all kinds of products) are distributive over addition. And the Frobenius product on a space of matrices is just a particular inner product.
Your formula doesn't really make sense, if the dot is an inner product, then the outcome is a scalar, and the parenthesised part on the left is subtracting a scalar from a non-square matrix, which cannot be done. It would make sense if you advanced the first opening parenthesis to just after the initial minus sign.