I have to think this is an actor faux-pas.
I cannot imagine someone so steeped in 'manners and etiquette', brought up to know Debrett's inside and out, to ever get this wrong.
Without going into too much detail on British cultural heritage and the class system, the pencil grip is a middle class affectation, a poor copy of the correct method, seemingly originating in some mistaken belief that it was 'more delicate' therefore 'more proper'.
It is found culturally amusing by both the upper and working classes, neither of whom would ever use this grip.
I managed to find one photo of Prince Philip holding a dinner/butter knife. I'm aware one photo doesn't prove absolutely that was how he always held it, but unless someone can find another picture with the wrong grip, I'm going to stick with my opinion.
As to why no-one noticed it during production. This form of 'correctness' along with many other post-Victorian ideas, is losing ground as younger people tend to neither know nor care for such outmoded 'manners'.
By the time anyone spotted it, it may have been too late or deemed too unimportant to set aside a day for reshoots… or that no-one wanted to point this out to the actor as being a 'dead-giveaway, middle-class affectation'.
I realise, as I'm sitting here eating breakfast at my desk, that as it's egg on toast I'm using the 'correct' grip for both utensils.
If it had been a meal that required no cutting, I'd just swap hands and use my fork like a ladle - I'm not terribly precious about it all ;)
I took up the challenge from steelersquirrel in comments below - find another instance he is seen to use the same grip. After quite some searching I got this, from Hysteria (2011). There are quite long scenes with the family eating, mainly with hands tantalisingly hidden behind other tableware or right on edge of frame, but I got this one clear shot where he is definitely using the same grip as 'Prince Philip'. It's a long enough shot to see him use this very distinctly. Oddly, towards the end of the shot you actually see him swap grip to the 'correct' method, right before a cut to another angle, using that correct grip again… then as we cut back to the first angle he is using the 'middle-class' grip once more. Continuity error ;)
Time stamp around 33:40 minutes if anyone want to check the whole scene