First, is he a special advisor to a government department or a policy advisor to either of the two parties in power?

No such thing as a government advisor. He is either ministerial / department advisor or a political advisor.

Any reporting of possible conflict of interest needs to be in respect of who he advises. An expert on fracking is an excellent departmental advisor, whilst politically, his vested interest could cloud his judgement.

Some advisors are about "shall we" whilst some are about "how do we." I'm not knocking the concern, just whether probity is an issue or not.