It does to some extent. When the bubbles rise, the spoon will cut the large pack of bubbles in two, making the center collapse, so there are now two packs of bubbles on each side thanmt can’t go as high because the shape (a half circle instead of a full circle) makes the building of bubbles less stable.
You don’t need a spoon or it to be wooden though (although the material may have a small influence).
Wooden spoons tend to be bulkier so maybe that’s why it’s always said to use a wooden one.
There are two possible effects from my point of view: the wood doesn’t heat the same way metal does, so it may have an impact on this ; and the wood is porous, so the water and stuff will not interact with the surface in the same way it would do with metal. But I doubt both those effects are that important.
As you say, maybe the thickness matters.
