Honestly, that FM radio antenna is a really crap antenna and you are thinking about a whole tier or 3 better design when you talk about a frequency matched dipole. I wouldn't be surprised if the length was unrelated to the frequency and just random. For receive only on signals as local and strong as FM radio it just doesn't matter much.
Yes, the way that verticle bazooka works is that the fold point changes the behavior of the wires in a way that causes it to act like a feed point on a dipole even though the core is continuous and the shield folds back on itself.
Antenna design is very much a case for the midwit meme with both ends being "just experiment and see how it works." Even people who are very knowledgeable are often surprised that a new design they encounter works at all.