"Person of colour", in my corner of the UK, is considered as bad as "coloured", for the same reason really: it "otherises" anyone who's not white, presuming that the world is made of "whites" and "others".
But really these things are not really logical in most cases. Ad es. blacks are blacks, but then you have "asians" who are not "east-asians" nor "south-asians", "latinos" that definitely did not originate from anywhere near the Italian peninsula, "white-irish" as different from "white-british", etc etc. In the inevitable public-authority forms, these days I'm often tempted to just tick "Mixed-other" - I'm whiter than daisies, but in my blood there are probably a dozen different "types of people" who invaded Italy through the centuries (africans, germans, etc etc).
Maybe I just don't get the social or cultural context, but this always confuses me. There's plenty of occasions where referring to one's skin color is relevant.
I'm light-skinned enough to be close to ginger, for instance, and this subject has both practical, conversational and occasionally humoristic value when the subject of being outside in the sun comes up. If I'm planning any sort of trip with someone who even has southern European ancestry, not to mention Middle-Eastern or African, it's hardly an unnatural subject to show up in passing.
From that context, it seems incredibly silly that the whole subject just be taboo and remain unmentioned.
It's generally not taboo when everyone is sure of each others intentions. E.g. I could if I wanted to safely tell my ex a borderline racist joke and she'd know there was no ill will intended, but if a stranger told her the same joke she'd be wondering what the intent behind it was.
Establishing that trust can sometimes take a while, but it tends to help a lot to simply ask about which terms someone prefers etc. if unsure, because just expressing a willingness to adjust tends to be a good sign.
I don't mean to speak for GP, but I'm also in the UK and pretty sure they meant the specific phrase, not mention of skin colour at all. In my experience 'black' here is as casual, normal, and inoffensive as 'white' or 'brown'. (Assuming it is relevant of course. If you're just arbitrarily chucking the description in all the time then yeah it doesn't matter what word or phrase you call it, it's still discriminatory. It'd be just as weird to do so with 'white'.)
If you mean re 'mixed other', I think the point was just that the categories don't apply well, and 'mixed' is technically true of essentially everyone so eh why not. (I believe 'white - other' is an option though, if that's applicable.)
But really these things are not really logical in most cases. Ad es. blacks are blacks, but then you have "asians" who are not "east-asians" nor "south-asians", "latinos" that definitely did not originate from anywhere near the Italian peninsula, "white-irish" as different from "white-british", etc etc. In the inevitable public-authority forms, these days I'm often tempted to just tick "Mixed-other" - I'm whiter than daisies, but in my blood there are probably a dozen different "types of people" who invaded Italy through the centuries (africans, germans, etc etc).