Uh… just gonna interject with my experience as a Mandarin speaker. Hope no one minds.
My understanding is that the fundamental, literal meaning of 格 is ‘box’ or ‘lattice’ (a box being one unit of the lattice), as in 格子, which is translated as ‘lattice’ by my EN-JP dictionary. In Mandarin, for instance, I would say that a sheet of grid paper is ‘paper that has 格子 on it’, and I might refer to each box (square/rectangle) in the grid as a 格 or 格子. In Japanese, I believe that 格子 can’t be used to designate a single box like in Mandarin, but the ‘lattice’ idea remains. And you’ll notice that one of the definitions of the kanji 格 is
③ 四角に組みあわせたもの。「格子（こうし）」「骨格」「格天井（ごうてんじよう）」
‘An object/objects put together into a square/quadrilateral’ (followed by examples)
so the idea clearly also exists in Japanese.
If it helps (though I don’t know if this is how the character came about), think about it like this:
In the old days, most civilisations would make lattices out of wood, right? (Other materials were too difficult/expensive to shape.) And the fundamental concept behind ‘each’ is ‘individualisation’, right? So 格 is in fact ‘wood that individualises’: in other words, a lattice.
The next step to understanding this kanji would be to link the literal meaning (‘lattice’, with the idea of ‘boxing things up’ or ‘compartmentalising’) to the meanings you’ve noticed. There are three main sorts: physical structures (like in 骨格), unique characteristics (like in 人格 and 性格) and standards (like in 資格). So here’s the question: what’s a lattice? It’s a framework made up of individual compartments, right? But a framework is at once 1. a physical structure and 2. (figuratively) a standard. That’s two of three meanings down! What about the last one? Well, compartmentalisation means each compartment can be unique, right? Plus, compartmentalisation is also about categorisation, which is putting things into a box, which is often what we think about in relation to 人格 and 性格, because everyone has a different character and personality type.
I hope that makes sense, and that my transition from the literal meaning to the figurative meaning didn’t seem too far-fetched. 
By the way, one final thing:
You’ve got the right idea, but I think the fundamental meaning of 性, as it’s used both Japanese and Chinese, is ‘essence’ i.e. a set of defining characteristics. For example, 活性 means ‘activity’ in the sense of ‘the condition of being active’, but it’s literally ‘the essence of being alive’. ‘Gender’ is just a special use of the kanji (maybe because those who first used the character had d tendency to see gender as something that defines us). Like you said, it’s about ‘the things that make you who you are’, or in the case of objects, it’s what makes certain objects what they are.
Whatever it is though, all the best with kanji! 頑張ってください！
Edit: I realise (upon thinking about your original post) that my explanation may not be in the spirit of WaniKani keywords and mnemonics, and it’s even more complicated given that I used the meaning of 各, which is taught after 格.
If I were to follow your model, I would propose ‘framework’ as the keyword for 格 and ‘boxy wood makes a framework’ (based on the shape of the 各 component) as a mnemonic.
I don’t know if other users would find this logical. I’m really just here to participate in discussions, and I haven’t tried the WaniKani programme yet. I wrote this post to help people understand what the kanji literally means, and how to connect that to other meanings. That’s all.