WINGS CAN ONLY BE A NUMBER ADJACENT TO YOUR TYPE.
I cannot tell you how many times I have to explain to someone that a wing is not the type you're second closest to in your quiz scores/personality.
I'm sure a lot of people on the forum have dealt with similar problems: you get someone proclaiming they're 9w6 or 3w7. Heh, for a while I know I thought I was 4w9.
I think this is because people confuse the point of the wing. They assume it's a "second-best," when, if that were true, we could have a 6w9 integrating towards 9...making the process entirely more complicated because, if we take the application of wings into account, you'd have the traits of a 9 anyway.* This is why the wing system works the way it does--to avoid those complications.
The enneagram types are in a certain order for a reason. We keep them so that the triads are bunched together:
Heart (2, 3, 4)
Thinking (5, 6, 7)
Instinct (8, 9, 1)
Certain enneagrams (4, 5, 7, 8, 1, 2) are settled right next to another triad, and are quite likely to have similar behavior to that other triad. Many people (from what I've noticed on this forum) relate to both their possible wings, although one is dominant. I noticed this especially in the 4w3 vs 4w5 discussion I started earlier.
For the enneagrams that are NOT near another triad (9s, 3s, and 6s), they are unique in that they disintegrate towards a a triad besides their own and then integrate towards the third. There's a series of checks and balances here; a 3 may only be able to have a Heart wing, but integrates into a 6 (Thinking) and disintegrates into a 9 (Instinct).
I'm too lazy to go into how the shape of the enneagram affects the Harmonics and the Hornevian groups, but basically: when someone tells you that the wing is the number adjacent to your type that you most identify with, we're not saying this for no reason. It's shaped the way it is to try and portray a precise system; if we can question how the wings work, then why don't we question how the integration and stress points work as well, and make it way too flexible to work for a general population.
I hope this was coherent, and any other explanations would be wonderful.
*(People also don't realize that a similar score on two types can make sense. I'll use myself as an example: I scored similarly on 4 and 9. I related to the 9 profile because I shy away from conflict and like there to be peace. After reading the profiles, though, I understood that those moods for me are caused by being a 4 disintegrating into 2 and seeking approval, not by being a 9 type. 9s are willing to sacrifice parts of themselves to keep the peace; a 4 is obsessed with being an individual and couldn't fathom any compromise with their identity. So a similar behavior can exist with different motivations.)