OK, the New Forest DC version is out of date. I found the current one on the Portsmouth Port web site:
It does show Isle of Wight District Council as a licensing authority. However it still says (and this is the wording that Pete7 alluded to):
"A licence under this section shall not be required for any boat or vessel duly licensed by or under any regulations of the Marine & Coastguard Agency (or for a person in charge of or navigating such a boat or vessel)."
Which I think makes it pretty clear that that the Local Authority licences are local alternatives to full MCA coding, not an additional requirement. I suppose it's possible that the Isle of Wight Council has some byelaw that applies, but I think that's pretty unlikely.
This assumes that it's the skipper of an MCA coded boat we're talking about, not someone wanting to use their existing qualifications instead of a Boatman's Certificate for a Local Authority registered boat. If that's the case then these are the exemptions:
"Persons holding a current MCA Certificate of Competency for Deck Officers, RYA Yacht master or Coastal Skipper Certificate, MCA Boat Master's Certificate or other higher qualifications shall be exempt from all parts of the examination except that some licensing authorities may wish to examine all applicants on local knowledge."
Either way the advice appears to be wrong!