tl;dr: Because wood is cheap and common, making it easy to introduce plot tension for otherwise overpowered characters.
Beyond that, wood is generally viewed as part of the Earth, a "force of nature" and representative of life, natural order, balance, etc. It invokes images of huge forests standing for thousands of years against opposing forces.
Or, depending on the tone of the work, it's such a seemingly silly weakness that it makes good comic relief.
First of all, since you're asking about very different science fiction universes, you are asking a fundamentally out-of-universe question, so the only real answer is going to be practical, and probably not very satisfying.
As mentioned in the answer you already cited, any time you have an overpowered character, you run the risk that the audience will lose any sense of drama or tension over that character's ability to get out of trouble. You need to introduce some weakness to keep things even. But you need to be careful what you select if you want it to be effective.
For any plot element to make a legitimate weakness, it needs to be something that your hero is reasonably likely to encounter, and that writers can introduce without jumping through a lot of hoops. (Contrast with, for example, Superman's weakness, and the amount of justification and effort it takes to explain why there's so much kryptonite on Earth.) You'll want to pick something that is easy enough to bring into the story, but not so common that your hero is constantly hamstring. This is why, for example, "air" or "water" or even "stone" might be a bad idea. Materials like wood strike a nice balance of being common while not being ubiquitous.
Besides just being one of the common materials you could pick, there are some specific reasons why wood seems to make a good foil. For starters, it's "all natural", so there is a sense of poetic justice that this super-advanced high-tech screwdriver can handle anything artificial we throw at it, but can't deal with a stick. Plus, in the Doctor Who case, it's just funny. (e.g. when the Doctor meets the creatures made of wood in the Christmas special and says something like "I knew this day would come.")
Wood is also something we are all familiar with; we understand where it comes from, how it works, what it does. It's simple, straightforward, and easy for even the most primitive society to work with. The idea that something so simple and basic can be a foil against impossibly powerful opposing weapons gives a sense of balance, that the "common man" isn't completely helpless in the face of these awesome forces. The ret-con justification for the Lantern ring's weakness formalizes this idea in-universe: the Guardians picked wood because it was the only source of weaponry available to primitive mankind, thus giving them a means to fight back against rogue Lanterns.