I think the adjoint you need is:
@adjoint Matrix(a::MyMat) = a.m, Δ -> (MyMat(Matrix(Δ), 0), )
I think what’s happening is that the default version for this doesn’t quite work with your matrix type and so it defaults to returning a named tuple (m=.., n=..) rather than a proper MyMat, but then the next step in the backpropagation tries to take the transpose of that named tuple which doesn’t work.
Btw, this is in some sense the same question I asked here (How to deal with Zygote sometimes "pirating" its own adjoints with worse ones?), there is a generic solution given there which might end up being helpful, since this sort of thing tends to happen alot in my experience with Zygote and custom matrix types.