I also don't think being at the bottom of a popularity list makes them unpopular. The least popular would be correct, but unpopular implies the majority of people don't like them, which isn't necessarily true. It would be like an election between two extremely favorable Prime Ministers (if only this was common), so someone has to win and someone has to lose, but that doesn't mean that even a single person disliked the loser.
This is the worst kind of semantics - not even really technically correct, and actually obfuscatory of the point.
The survey from D&D beyond is fundamentally flawed. Most notably, the free classes are leading most likely due to them being free and available to all players.
The survey accounted for this in its methodology, which they discussed at quite some length at the time. The fact that you are unaware of this means your criticism is one from ignorance, and it is invalid because this was accounted for. If you want to look into the methodology, you can go dig up the threads from the time. I made a similar assumption back then, I believe, but I was wrong.
But also, it could possibly be because the Purple Dragon Knight doesn't even have an identity. The Champion is someone that dominates with sheer will, like an arena champion. The battlemaster is tactical and understands their enemy's weakness, like someone who has mastered battle. The eldritch knight has mastered the arcane, much like a magical knight that chose magic in a way other than religion or study. The Purple Dragon Knight...what in the world does that name even mean? It's a knight, sure, but um...purple dragon? That doesn't even exist in D&D lore and it's meaningless. You aren't replicating a Dragon.
Everyone knows what a cavalier is unless you don't know the definition of the word. I know the definition of the words purple and dragon and knight and I have no clue what they are supposed to mean when put together.
This is totally irrelevant, given the Bladesinger subclass is in the number one subclass for Wizards (albeit narrowly). All the same criticisms apply to the Bladesinger's name perhaps even more strongly.
Furthermore, the Bladesinger was in the same book as the PDK, so has the same level of people able to access it, so that also negates any arguments there.
The PDK is less popular than the equally ridiculously-named Echo Knight, which comes from a book which has sold far fewer copies (AFAIK) than Sword Coast, and had only been out like a month when the survey we're discussing landed. Compare that to the Bladesinger, and it's pretty obvious that people are actually looking at the abilities and themes of subclasses. Looking at the popular subclasses they're pretty much all either ones with a strong theme (sometimes a bit edgelord-y) or strong mechanics, or those with both.
(As a semantic aside, I can't speak worldwide, but if you say "cavalier" in the UK, people think "perfumed fop in a fancy hat on a horse who lost a war due to ineptitude", not "armoured knight" - I strongly suspect that the number of English-speakers who think of a D&D-style cavalier when they hear the term is, well, more limited than you think.)